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Volume of a cylinder

Here you will learn about the volume of a cylinder, including how to calculate the volume of a cylinder given its radius, area and perpendicular height.

Students will first learn about volume of cylinders as part of geometry in grade 7 and again in grade 8 and high school geometry.

What is the volume of a cylinder?

The volume of a cylinder is the amount of space there is inside a cylinder.

In order to find the volume of a cylinder you first need to find the circular area of the base.

To do this, you can use the formula for calculating the area of a circle, which is

Then multiply the area of the circular base by the height (or length) of the cylinder.

To do this, you can use the formula for the volume of a cylinder , which is

Where \textbf{r} is the radius of a cylinder and \textbf{h} is the perpendicular height of a cylinder.

For example, find the volume of this cylinder with radius of the base 7~cm and perpendicular height 10~cm.

Common Core State Standards

How does this relate to 7 th grade and high school math?

• Grade 7 – Geometry (7.G.B.6) Solve real-world and mathematical problems involving area, volume and total surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
• High School – Geometry (HSG.GMD.A.3) Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

[FREE] Volume Check for Understanding Quiz (Grade 6 to 8)

Use this quiz to check your grade 6 to 8 students’ understanding of volume. 10+ questions with answers covering a range of 6th, 7th and 8th grade volume topics to identify areas of strength and support!

How to calculate the volume of a cylinder

• Write down the formula: \text{Volume} = \pi r^2 h

Substitute the given values.

Work out the calculation.

Write the final answer, including units.

Volume of a cylinder examples

Example 1: volume of a cylinder.

Find the volume of the cylinder with radius 3~cm and perpendicular height 5~cm.

Give your answer to 1 decimal place.

Write down the formula.

To answer the question you need the formula for the volume of a cylinder.

2 Substitute the given values.

Substitute the value of the radius r and the perpendicular height h into the formula.

3 Work out the calculation.

Use a calculator to work out the volume.

4 Write the final answer, including units.

Check what form the final answer needs to be written in. You may need to leave your answer in terms of \pi or as a decimal with rounding. Here you are asked to give the answer to 1 decimal place.

The volume of the cylinder is: 141.4 \mathrm{~cm}^3.

Example 2: volume of a cylinder

Find the volume of the cylinder with radius 4.8~cm and perpendicular height 7.9~cm.

Give your answer to 3 significant figures.

\text { Volume }=\pi r^2 h

V=\pi \times 4.8^2 \times 7.9

Check what form the final answer needs to be.

You may need to leave your answer in terms of \pi or as a decimal with rounding.

Here you are asked to give the answer to 3 significant figures.

V=571.820=572 \mathrm{~cm}^3

The volume of the cylinder is: 572 \mathrm{~cm}^3.

Example 3: volume of a cylinder

Find the volume of the cylinder with radius 3~cm and perpendicular height 7~cm.

Leave your answer in terms of \pi.

Write down the formula .

V=\pi \times 3^2 \times 7

Work out the volume. (Focusing on the number parts of the calculation).

Check what form the final answer needs to be written in. You may need to leave your answer in terms of \pi or as a decimal with rounding.

V=63 \pi=63 \pi \mathrm{~cm}^3

The volume of the cylinder is: 63 \pi \mathrm{~cm}^3.

Example 4: volume of a cylinder

Find the volume of the cylinder with radius 4~cm and perpendicular height 10~cm.

V=\pi \times 4^2 \times 10

V=160 \pi=160 \pi \mathrm{~cm}^3

The volume of the cylinder is: 160 \pi \mathrm{~cm}^3.

Example 5: using the formula to find a length

The volume of a cylinder is 1600~cm^3. Its radius is 9~cm. Find its perpendicular height. Give your answer to 2 decimal places.

The question involves the volume of a cylinder, so you need the formula for the volume of a cylinder.

V=\pi r^2 h

You are given the volume and the radius, so substitute these into the formula.

1600=\pi \times 9^2 \times h

You need to rearrange the formula to find the value of h.

1600=\pi \times 81 \times h

\cfrac{1600}{\pi \times 81}=h

Here you are asked to give the answer to 2 decimal places.

h=6.287=6.29 \mathrm{~cm}

The perpendicular height of the cylinder is: 6.29~cm.

Example 6: using the formula to find a length

The volume of a cylinder is 1400~cm^3. Its perpendicular height is 15~cm. Find its radius. Give your answer to 2 decimal places.

You are given the volume and the perpendicular height, so substitute these into the formula.

1400=\pi \times r^2 \times 15

\cfrac{1400}{\pi \times 15}=r^2

r^2=29.70892

r=\sqrt{29.70892}

Check what form the final answer needs to be written in.

r=\sqrt{29.70892}=5.4505=5.45 \mathrm{~cm}

The radius of the cylinder is: 5.45~cm.

Teaching tips for volume of a cylinder

• Implement engaging visual aids and real-world examples to illustrate how the formula applies, fostering a deeper comprehension of 3d shapes. Using visual aids will also help English Language Learners comprehend the complexity of the three dimensional space and three-dimensional shapes.
• Incorporate interactive activities and group exercises that encourage hands-on exploration of cylinder volume calculations, ensuring students actively participate in the learning process. Incorporating open-ended tasks for students to find the surface area of a cylinder, cuboid, or oblique cylinder as well as a focus on the area of the cylinder and base of the cylinder will scaffold student learning appropriately.
• Provide step-by-step problem-solving tutorials, addressing common challenges students might encounter when finding the volume of a cylinder, thereby enhancing the accessibility and usefulness of your educational content.
• Highlight the importance of exposing students to diverse cubic units such as cubic inches, cubic centimeters, cubic meters, and cubic feet, as it not only reinforces the concept of volume but also fosters a versatile and practical comprehension of three-dimensional space.

Easy mistakes to make

• Incorrect radius calculation When finding the volume of a cylinder, be cautious not to miscalculate the radius, as errors in this fundamental measurement can lead to inaccuracies in the final volume. Emphasize the importance of doubling the radius squared and verify calculations to prevent mistakes.
• Misinterpreting height measurements Avoid the mistake of misinterpreting or inaccurately measuring the height of the cylinder. Stress the need for precision in obtaining height values, and remind students to double-check measurements to ensure accuracy in volume calculations.
• Omitting units in the final answer In the excitement of solving volume problems, it’s easy to overlook including the appropriate units in the final answer. Remind students to always attach the correct cubic units (cubic inches, cubic centimeters, cubic feet, cubic meters, etc.) to their volume results, enhancing clarity and correctness.
• Neglecting the order of operations Students may inadvertently neglect the proper order of operations when applying the volume formula. Reinforce the importance of parentheses and guiding students to follow the correct sequence to prevent errors in their calculations.
• Overlooking the \textbf{P}\,\textbf{i} value Students may forget to incorporate the value of Pi accurately, leading to significant errors in volume calculations. Stress the consistent use of the correct \pi value and caution against rounding too early in the calculation process to maintain precision in results.

Related volume lessons

• Volume of a rectangular prism
• Volume of a hemisphere
• Volume of a prism
• Volume of a sphere
• Volume of a cone
• Volume of a cube
• Volume of a triangular prism
• Volume of a pyramid
• Volume of square pyramid
• Volume formula

Practice volume of a cylinder questions

1) Find the volume of a cylinder with a radius 3.2~cm and perpendicular height 9.1~cm. Give your answer to 3 significant figures.

You are finding the volume of a cylinder so substitute the value of r and h into the formula.

2) Find the volume of a cylinder with a radius 5.3~cm and perpendicular height 3.8~cm. Give your answer to 3 significant figures.

3) Find the volume of a cylinder with a radius 8~cm and perpendicular height 7~cm. Leave your answer in terms of \pi.

4) Find the volume of a cylinder with a radius 4~cm and perpendicular height 8~cm. Give your answer to 3 significant figures.

5) The volume of a cylinder is 250~cm^3. Its radius is 2.9~cm. Find its perpendicular height. Give your answer to 2 decimal places.

Using the formula, substitute the value of the volume and the value of the radius and rearrange to find the radius.

6) The volume of a cylinder is 800~cm^3. Its perpendicular height is 9.2~cm. Find its radius. Give your answer to 2 decimal places.

Using the formula, substitute the value of the volume and the value of the perpendicular height and rearrange to find the radius.

Volume of a cylinder FAQs

Common mistakes include misinterpreting height measurements, miscalculating the radius, and omitting units. Learn to recognize and avoid these errors for accurate volume calculations. Additionally, double-check your calculations and ensure you follow the correct order of operations.

Yes, you can use various cubic units, such as cubic inches or cubic centimeters, when calculating cylinder volume. Understanding different units provides versatility in solving real-world problems. Be consistent with your unit usage to maintain precision in your answers.

Ensure accuracy by consistently using the appropriate Pi value throughout your calculations. Maintain precision by avoiding premature rounding in the process. Remember to keep \pi in your calculations until the final step for accurate results.

Engage in hands-on activities like measuring the volume of cylindrical objects such as cans or jars. These real-world examples enhance your grasp of the volume formula and its applications. Experiment with various objects to reinforce the concept and its practical implications in different scenarios.

The next lessons are

• Surface area
• Pythagorean theorem
• Trigonometry

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Volume of Cylinders

A cylinder is a solid with two congruent circles joined by a curved surface.

In the above figure, the radius of the circular base is r and the height is h.

The volume of the cylinder is the area of the base × height. Since the base is a circle and the area of a circle is πr 2 then the volume of the cylinder is πr 2 × h.

Volume of cylinder = πr 2 h

Surface Area of cylinder = 2πr 2 + 2πrh

Calculate the volume of a cylinder where:

a) the area of the base is 30 cm 2 and the height is 6 cm. b) the radius of the base is 14 cm and the height is 10 cm.

Sometimes you may be required to calculate the volume of a hollow cylinder or tube or pipe.

Volume of hollow cylinder: = πR 2 h – πr 2 h = πh (R 2 – r 2 )

The figure shows a section of a metal pipe. Given the internal radius of the pipe is 2 cm, the external radius is 2.4 cm and the length of the pipe is 10 cm. Find the volume of the metal used.

Solution: The cross section of the pipe is a ring: Area of ring = [ π (2.4) 2 – π (2) 2 ]= 1.76 π cm 2

These videos show how to solve word problems about cylinders.

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Volume of Cylinder

The volume of a cylinder is the capacity of the cylinder which calculates the amount of material quantity it can hold. In geometry, there is a specific formula to calculate the volume of a cylinder that is used to measure how much amount of any quantity whether liquid or solid can be immersed in it uniformly. A cylinder is a three-dimensional shape with two congruent and parallel identical bases. There are different types of cylinders. They are:

• Right circular cylinder: A cylinder whose bases are circles and each line segment that is a part of the lateral curved surface is perpendicular to the bases.
• Oblique Cylinder: A cylinder whose sides lean over the base at an angle that is not equal to a right angle.
• Elliptic Cylinder: A cylinder whose bases are ellipses.
• Right circular hollow cylinder: A cylinder that consists of two right circular cylinders bounded one inside the other.

The formula to find the volume of a cylinder is V = πr 2 h. Let us learn more about this formula in the upcoming sections.

What is the Volume of a Cylinder?

The volume of a cylinder is the number of unit cubes (cubes of unit length) that can be fit into it. It is the space occupied by the cylinder as the volume of any three-dimensional shape is the space occupied by it. The volume of a cylinder is measured in cubic units such as cm 3 , m 3 , in 3 , etc. Let us see the formula used to calculate the volume of a cylinder.

Definition of a Cylinder

A cylinder is a three-dimensional solid shape that consists of two parallel bases linked by a curved surface. These bases are like a circular disk in a shape . The line passing from the center or joining the centers of two circular bases is called the axis of the cylinder.

Volume of Cylinder Formula

We know that a cylinder resembles a prism (but note that a cylinder is not a prism as it has a curved side face), we use the same formula of volume of a prism to calculate the volume of a cylinder as well. We know that the volume of a prism is calculated using the formula,

V = A × h, where

• A = area of the base

Using this formula, the formulas of volume of cylinder are:

• The formula for volume of a right circular cylinder is, V = πr 2 h (r = radius, h = height)
• The formula for volume of an oblique cylinder is, V = πr 2 h (r = radius, h = height)
• The formula for volume of an elliptic cylinder is, V = πabh (a and b = radii, h = height)
• The formula for volume of a right circular hollow cylinder is, V = π(R 2 - r 2 )h (R = outer radius, r = inner radius, h = height)

Now we will apply the formula V = A × h to calculate the volume of different types of cylinders.

Volume of a Right Circular Cylinder Formula

We know that the base of a right circular cylinder is a circle and the area of a circle of radius 'r' is πr 2 . Thus, the volume (V) of a right circular cylinder, using the above formula (V = A × h), is,

• 'r' is the radius of the base (circle) of the cylinder
• 'h' is the height of the cylinder
• π is a constant whose value is either 22/7 (or) 3.142.

Thus, the volume of cylinder directly varies with its height and directly varies with the square of its radius. i.e., if the radius of the cylinder becomes double, then its volume becomes four times.

Formula to Find Volume of an Oblique Cylinder

The formula to calculate the volume of cylinder (oblique) is the same as that of a right circular cylinder. Thus, the volume (V) of an oblique cylinder whose base radius is 'r' and whose height is 'h' is,

Formula to Calculate Volume of an Elliptic Cylinder

We know that an ellipse has two radii. Also, we know that the area of an ellipse whose radii are 'a' and 'b' is πab. Thus, the volume of an elliptic cylinder is,

• 'a' and 'b' are the radii of the base (ellipse) of the cylinder.
• 'h' is the height of the cylinder.

Volume of a Right Circular Hollow Cylinder Formula

As a right circular hollow cylinder is a cylinder that consists of two right circular cylinders bounded one inside the other, its volume is obtained by subtracting the volume of the inside cylinder from that of the outside cylinder. Thus, the volume (V) of a right circular hollow cylinder is,

V = π(R 2 - r 2 )h

• 'R' is the base radius of the outside cylinder.
• 'r' is the base radius of the inside cylinder.
• π is a constant whose value is 22/7 (or) 3.142.

How To Find the Volume of Cylinder?

Here are the steps to find the volume of cylinder:

• Identify the radius to be 'r' and height to be 'h' and make sure that they both are of the same units.
• Substitute the values in the volume formula V = πr 2 h.
• Write the units as cubic units.

Example: Find the volume of a right circular cylinder of radius 50 cm and height 1 meter. Use π = 3.142.

The radius of the cylinder is, r = 50 cm.

Its height is, h = 1 meter = 100 cm.

Its volume is, V = πr 2 h = (3.142)(50) 2 (100) = 785,500 cm 3 .

Note: We need to use the formula to find the volume of a cylinder depending on its type as we discussed in the previous section. Also, assume that a cylinder is a right circular cylinder if there is no type given and apply the volume of a cylinder formula to be V = πr 2 h.

Important Notes on Volume of Cylinder:

• The volume of a cylinder is calculated using the formula, V = πr 2 h, where r is the radius of its circular base and 'h' is the perpendicular distance (height) between the centres of the bases.
• If diameter (d) is given, then find the radius (r) using r = d/2 and then substitute in the above formula to find the volume of cylinder.

Volume of Cylinder Examples

Example 1: Find the volume of a cylindrical water tank in litres whose base radius is 25 m and whose height is 120 m. Use π = 3.14.

The radius of the cylindrical tank is, r = 25 m.

Its height is, h = 120 m.

Using the formula of volume of cylinder, the volume of the tank is,

V = (3.14)(25) 2 (120) = 235500 cubic meters.

The volume of a cylinder in litres is obtained by using the conversion formula 1 cubic meter = 1000 liters.

Thus, the volume of the tank in liters is: 235500 × 1000 = 235,500,000

Answer: The volume of the given cylindrical tank is 235,500,000 liters.

Example 2: Calculate the volume of an elliptic cylinder whose base radii are 7 inches and 10 inches, and whose height is 15 inches. Use π = 22/7.

The base radii of the given elliptic cylinder are,

a = 7 inches and b = 10 inches.

Its height is, h = 15 inches.

Using the volume of cylinder formula, the volume of the given elliptic cylinder is,

V = (22/7) × 7 × 10 × 15 = 3300 cubic inches.

Answer: The volume of the given cylinder is 3,300 cubic inches.

Example 3: What is the volume of the cylinder with a radius of 4 units and a height of 6 units?

Since the exact type of cylinder is not mentioned, we need to assume it is a right circular cylinder.

Radius,r = 4 units and height,h = 6 units

Volume of the cylinder, V = πr 2 h cubic units.

V = (22/7) × (4) 2 × 6 V = 22/7 × 16 × 6

V= 301.71 cubic units.

Answer: The volume of the cylinder is 301.71 cubic units.

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Practice Questions on Volume of Cylinder

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FAQs on Volume of Cylinder

What is the meaning of volume of cylinder.

The volume of a cylinder is the amount of space in it. It can be obtained by multiplying its base area by its height. The formula to find the  volume of a cylinder of base radius 'r' and height 'h' is V = πr 2 h.

What Is the Formula for Calculating the Volume of a Cylinder?

The formula for calculating the volume of a cylinder is V = πr 2 h, where

• 'r' is the radius of the base of the cylinder
• π is a constant whose value is equal to approximately 3.142.

What is the Volume of a Cylinder with Diameter?

Let us consider a cylinder of radius 'r', diameter 'd', and height 'h'. The volume of a cylinder of base radius 'r' and height 'h' is V = πr 2 h. We know that r = d/2. By substituting this in the above formula, V = πd 2 h/4.

What Is the Ratio of the Volume of a Cylinder and a Cone?

Let us consider a cylinder and a cone , each with base radius 'r' and height 'h'. We know that the volume of the cylinder is πr 2 h and the volume of the cone is 1/3 πr 2 h. Thus the required ratio is 1:(1/3) (or) 3:1.

How To Calculate Volume of a Cylinder With Diameter and Height?

The volume of a cylinder with base radius 'r' and height 'h' is, V = πr 2 h. If its base diameter is d, then we have d = r/2. Substituting this in the above formula, we get V = πd 2 h/4. Thus, the formula to find the volume of a cylinder with the diameter (d) and height(h) is V = πd 2 h/4.

How To Find Volume of Cylinder With Circumference and Height?

We know that the circumference of a circle of radius 'r' is C = 2πr. Thus, when the circumference of the base of a cylinder (C) and its height (h) are given, then we first solve the equation C = 2πr for 'r' and then we apply the volume of a cylinder formula, which is, V = πr 2 h.

How To Calculate Volume of Cylinder in Litres?

We can use the following conversion formulas to convert the volume of cylinder from m 3 (or) cm 3 to liters.

• 1 m 3 = 1000 liters
• 1 cm 3 = 1 ml (or) 0.001 liters
• Metric Conversion
• Unit Conversion

What Happens to the Volume of Cylinder When Its Radius Is Halved?

The volume of cylinder varies directly with the square of its radius. Thus, when its radius is halved, the volume becomes 1/4 th .

What Happens to the Volume of Cylinder When Its Radius Is Doubled?

We know that the volume of cylinder is directly proportional to the square of its radius. Thus, when its radius is doubled, the volume becomes four times.

How Do You Find Volume of Cylinder Using Calculator?

Volume of a cylinder calculator is a machine to calculate a cylinder's volume. To calculate the volume of a cylinder using a calculator we need to provide necessary inputs to the calculator tool, such as required dimensions like radius, diameter, height, etc. Try now the volume of a cylinder calculator enter the radius and height of the cylinder in the given box of the volume of a cylinder calculator. Click on the "Calculate" button to find the volume of a cylinder. By clicking the "Reset" button you can easily clear the previously entered data and find the volume of a cylinder for different values.

☛Also Check:

• Cylinder Calculator
• Surface Area of Cylinder Calculator
• Height of a Cylinder Calculator

What is the Area and Volume of a Cylinder?

The surface area of a cylinder is the total area or region covered by the surface of the cylinder. The surface area of a cylinder is given by two following formulas:

• The curved surface area of cylinder = 2πrh
• The total surface area of the cylinder = 2πr 2 +2πrh = 2πr(h+r)

The area of a cylinder is expressed in square units, like m 2 , in 2 , cm 2 , yd 2 , etc.

The volume of a cylinder is the total amount of capacity immersed in a cylinder that can be calculated using the volume of cylinder equation is V = πr 2 h and it is measured in cubic units.

• Surface Area of Cylinder Worksheets
• Volume of a Cylinder Worksheets
• Surface Area Formulas

How Does the Volume of a Hollow Cylinder Change When the Height is Doubled?

The volume of a hollow cylinder formula is V = π(R 2 - r 2 )h cubic units. According to the volume formula, we can see that volume is directly proportional to the height of the hollow cylinder. Therefore, the volume gets doubled when the height of the hollow cylinder is doubled.

What is the Volume of Cylinder in Terms of Pi?

The volume of cylinder is defined as the capacity of a cylinder which is indicated in terms of pi. The volume of a cylinder in terms of pi is expressed in cubic units where units can be m 3 , cm 3 , in 3 , ft 3 , etc.

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Volume Problem Solving

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To solve problems on this page, you should be familiar with the following: Volume - Cuboid Volume - Sphere Volume - Cylinder Volume - Pyramid

This wiki includes several problems motivated to enhance problem-solving skills. Before getting started, recall the following formulas:

• Volume of sphere with radius \(r:\) \( \frac43 \pi r^3 \)
• Volume of cube with side length \(L:\) \( L^3 \)
• Volume of cone with radius \(r\) and height \(h:\) \( \frac13\pi r^2h \)
• Volume of cylinder with radius \(r\) and height \(h:\) \( \pi r^2h\)
• Volume of a cuboid with length \(l\), breadth \(b\), and height \(h:\) \(lbh\)

Volume Problem Solving - Basic

Volume - problem solving - intermediate, volume problem solving - advanced.

This section revolves around the basic understanding of volume and using the formulas for finding the volume. A couple of examples are followed by several problems to try.

Find the volume of a cube of side length \(10\text{ cm}\). \[\begin{align} (\text {Volume of a cube}) & = {(\text {Side length}})^{3}\\ & = {10}^{3}\\ & = 1000 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

Find the volume of a cuboid of length \(10\text{ cm}\), breadth \(8\text{ cm}\). and height \(6\text{ cm}\). \[\begin{align} (\text {Area of a cuboid}) & = l × b × h\\ & = 10 × 8 × 6\\ & = 480 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

I made a large ice cream cone of a composite shape of a cone and a hemisphere. If the height of the cone is 10 and the diameter of both the cone and the hemisphere is 6, what is the volume of this ice cream cone? The volume of the composite figure is the sum of the volume of the cone and the volume of the hemisphere. Recall the formulas for the following two volumes: \( V_{\text{cone}} = \frac13 \pi r^2 h\) and \( V_{\text{sphere}} =\frac43 \pi r^3 \). Since the volume of a hemisphere is half the volume of a a sphere of the same radius, the total volume for this problem is \[\frac13 \pi r^2 h + \frac12 \cdot \frac43 \pi r^3. \] With height \(h =10\), and diameter \(d = 6\) or radius \(r = \frac d2 = 3 \), the total volume is \(48\pi. \ _\square \)

Find the volume of a cone having slant height \(17\text{ cm}\) and radius of the base \(15\text{ cm}\). Let \(h\) denote the height of the cone, then \[\begin{align} (\text{slant height}) &=\sqrt {h^2 + r^2}\\ 17&= \sqrt {h^2 + 15^2}\\ 289&= h^2 + 225\\ h^2&=64\\ h& = 8. \end{align}\] Since the formula for the volume of a cone is \(\dfrac {1}{3} ×\pi ×r^2×h\), the volume of the cone is \[ \frac {1}{3}×3.14× 225 × 8= 1884 ~\big(\text{cm}^{2}\big). \ _\square\]

Find the volume of the following figure which depicts a cone and an hemisphere, up to \(2\) decimal places. In this figure, the shape of the base of the cone is circular and the whole flat part of the hemisphere exactly coincides with the base of the cone (in other words, the base of the cone and the flat part of the hemisphere are the same). Use \(\pi=\frac{22}{7}.\) \[\begin{align} (\text{Volume of cone}) & = \dfrac {1}{3} \pi r^2 h\\ & = \dfrac {1 × 22 × 36 × 8}{3 × 7}\\ & = \dfrac {6336}{21} = 301.71 \\\\ (\text{Volume of hemisphere}) & = \dfrac {2}{3} \pi r^3\\ & = \dfrac {2 × 22 × 216}{3 × 7}\\ & = \dfrac {9504}{21} = 452.57 \\\\ (\text{Total volume of figure}) & = (301.71 + 452.57) \\ & = 754.28.\ _\square \end{align} \]

Try the following problems.

Find the volume (in \(\text{cm}^3\)) of a cube of side length \(5\text{ cm} \).

A spherical balloon is inflated until its volume becomes 27 times its original volume. Which of the following is true?

Bob has a pipe with a diameter of \(\frac { 6 }{ \sqrt { \pi } }\text{ cm} \) and a length of \(3\text{ m}\). How much water could be in this pipe at any one time, in \(\text{cm}^3?\)

What is the volume of the octahedron inside this \(8 \text{ in}^3\) cube?

A sector with radius \(10\text{ cm}\) and central angle \(45^\circ\) is to be made into a right circular cone. Find the volume of the cone.

\[\] Details and Assumptions:

• The arc length of the sector is equal to the circumference of the base of the cone.

Three identical tanks are shown above. The spheres in a given tank are the same size and packed wall-to-wall. If the tanks are filled to the top with water, then which tank would contain the most water?

A chocolate shop sells its products in 3 different shapes: a cylindrical bar, a spherical ball, and a cone. These 3 shapes are of the same height and radius, as shown in the picture. Which of these choices would give you the most chocolate?

\[\text{ I. A full cylindrical bar } \hspace{.4cm} \text{ or } \hspace{.45cm} \text{ II. A ball plus a cone }\]

How many cubes measuring 2 units on one side must be added to a cube measuring 8 units on one side to form a cube measuring 12 units on one side?

This section involves a deeper understanding of volume and the formulas to find the volume. Here are a couple of worked out examples followed by several "Try It Yourself" problems:

\(12\) spheres of the same size are made from melting a solid cylinder of \(16\text{ cm}\) diameter and \(2\text{ cm}\) height. Find the diameter of each sphere. Use \(\pi=\frac{22}{7}.\) The volume of the cylinder is \[\pi× r^2 × h = \frac {22×8^2×2}{7}= \frac {2816}{7}.\] Let the radius of each sphere be \(r\text{ cm}.\) Then the volume of each sphere in \(\text{cm}^3\) is \[\dfrac {4×22×r^3}{3×7} = \dfrac{88×r^3}{21}.\] Since the number of spheres is \(\frac {\text{Volume of cylinder}}{\text {Volume of 1 sphere}},\) \[\begin{align} 12 &= \dfrac{2816×21}{7×88×r^3}\\ &= \dfrac {96}{r^3}\\ r^3 &= \dfrac {96}{12}\\ &= 8\\ \Rightarrow r &= 2. \end{align}\] Therefore, the diameter of each sphere is \[2\times r = 2\times 2 = 4 ~(\text{cm}). \ _\square\]

Find the volume of a hemispherical shell whose outer radius is \(7\text{ cm}\) and inner radius is \(3\text{ cm}\), up to \(2\) decimal places. We have \[\begin{align} (\text {Volume of inner hemisphere}) & = \dfrac{1}{2} × \dfrac{4}{3} × \pi × R^3\\ & = \dfrac {1 × 4 × 22 × 27}{2 × 3 × 7}\\ & = \dfrac {396}{7}\\ & = 56.57 ~\big(\text{cm}^{3}\big) \\\\ (\text {Volume of outer hemisphere}) & = \dfrac{1}{2} × \dfrac{4}{3} × \pi × r^3\\ & = \dfrac {1 × 4 × 22 × 343}{2 × 3 × 7}\\ & = \dfrac {2156}{7}\\ & = 718.66 ~\big(\text{cm}^{3}\big) \\\\ (\text{Volume of hemispherical shell}) & = (\text{V. of outer hemisphere}) - (\text{V. of inner hemisphere})\\ & = 718.66 - 56.57 \\ & = 662.09 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

A student did an experiment using a cone, a sphere, and a cylinder each having the same radius and height. He started with the cylinder full of liquid and then poured it into the cone until the cone was full. Then, he began pouring the remaining liquid from the cylinder into the sphere. What was the result which he observed?

There are two identical right circular cones each of height \(2\text{ cm}.\) They are placed vertically, with their apex pointing downwards, and one cone is vertically above the other. At the start, the upper cone is full of water and the lower cone is empty.

Water drips down through a hole in the apex of the upper cone into the lower cone. When the height of water in the upper cone is \(1\text{ cm},\) what is the height of water in the lower cone (in \(\text{cm}\))?

On each face of a cuboid, the sum of its perimeter and its area is written. The numbers recorded this way are 16, 24, and 31, each written on a pair of opposite sides of the cuboid. The volume of the cuboid lies between \(\text{__________}.\)

A cube rests inside a sphere such that each vertex touches the sphere. The radius of the sphere is \(6 \text{ cm}.\) Determine the volume of the cube.

If the volume of the cube can be expressed in the form of \(a\sqrt{3} \text{ cm}^{3}\), find the value of \(a\).

A sphere has volume \(x \text{ m}^3 \) and surface area \(x \text{ m}^2 \). Keeping its diameter as body diagonal, a cube is made which has volume \(a \text{ m}^3 \) and surface area \(b \text{ m}^2 \). What is the ratio \(a:b?\)

Consider a glass in the shape of an inverted truncated right cone (i.e. frustrum). The radius of the base is 4, the radius of the top is 9, and the height is 7. There is enough water in the glass such that when it is tilted the water reaches from the tip of the base to the edge of the top. The proportion of the water in the cup as a ratio of the cup's volume can be expressed as the fraction \( \frac{m}{n} \), for relatively prime integers \(m\) and \(n\). Compute \(m+n\).

The square-based pyramid A is inscribed within a cube while the tetrahedral pyramid B has its sides equal to the square's diagonal (red) as shown.

Which pyramid has more volume?

Please remember this section contains highly advanced problems of volume. Here it goes:

Cube \(ABCDEFGH\), labeled as shown above, has edge length \(1\) and is cut by a plane passing through vertex \(D\) and the midpoints \(M\) and \(N\) of \(\overline{AB}\) and \(\overline{CG}\) respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p+q\).

If the American NFL regulation football

has a tip-to-tip length of \(11\) inches and a largest round circumference of \(22\) in the middle, then the volume of the American football is \(\text{____________}.\)

Note: The American NFL regulation football is not an ellipsoid. The long cross-section consists of two circular arcs meeting at the tips. Don't use the volume formula for an ellipsoid.

Answer is in cubic inches.

Consider a solid formed by the intersection of three orthogonal cylinders, each of diameter \( D = 10 \).

What is the volume of this solid?

Consider a tetrahedron with side lengths \(2, 3, 3, 4, 5, 5\). The largest possible volume of this tetrahedron has the form \( \frac {a \sqrt{b}}{c}\), where \(b\) is an integer that's not divisible by the square of any prime, \(a\) and \(c\) are positive, coprime integers. What is the value of \(a+b+c\)?

Let there be a solid characterized by the equation \[{ \left( \frac { x }{ a } \right) }^{ 2.5 }+{ \left( \frac { y }{ b } \right) }^{ 2.5 } + { \left( \frac { z }{ c } \right) }^{ 2.5 }<1.\]

Calculate the volume of this solid if \(a = b =2\) and \(c = 3\).

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Volume of a cylinder

The volume of a cylinder is the amount of space it encloses. Below is a paint bucket in the shape of cylinder. The volume of the cylinder is the maximum amount of paint the bucket can hold.

Formula for the volume of a cylinder

The formula for the volume, V, of a cylinder is:

V = π r 2 h

where r is radius of the base and h is the height of the cylinder.

If the base area, B, of the cylinder is given, the volume is:

The base radius of a cylinder is 2 and the height of the cylinder is 3. What is the volume of the cylinder?

V = π × 2 2 × 3 = 12π

Volume of a tube

A tube is a hollow cylinder, like the one shown below. The volume of a tube is:

V = πh(R 2 - r 2 )

where R is the outer radius, r is the inner radius, and h is the length of the tube.

The volume of a tube can be used to determine how much material is needed to construct shapes such as a hose or a pipe.

A cement drainage pipe has a length of 96 inches, outer diameter of 28 inches, and inner diameter of 24 inches. What is the volume of the pipe (how much cement do we need to build the pipe)?

The radius of the outer cylinder is R = 14 and the inner radius is r = 12. The height of both cylinders is 96 so the volume of the pipe is

V = π × 96 × (14 2 - 12 2 ) = 4992π ≈ 15,682.83

Therefore, the cement needed to build the pipe is 15,682.83 cubic inches

Volume of a substance in a partially filled cylinder

Volume in an upright cylinder.

When the cylinder is upright, or resting on one of its bases, we can find the volume of a substance in the cylinder as long as we know the height the substance reaches in the cylinder. The formula is the same as that for the volume of a right cylinder:

V = π r 2 h 1

where r is radius of the base and h 1 is the height the substance reaches in the cylinder.

A water tank in the shape of a cylinder has a radius of 5 and a height of 9. If the tank is two-thirds full, how much water does it contain?

Since the tank is two-thirds full, the height, h 1 , of the water level in the tank is 2/3 of the height of the tank:

h 1 = 2/3 × 9 = 6

Plugging this into the volume formula:

V = π × 5 2 × 6 = 150π

Volume in a cylinder resting on its side

The volume of a substance in a cylinder resting on its lateral surface can be found using the following formula:

where r is the radius of the base of the cylinder, h is the height that the substance reaches in the cylinder, and l is the length between the bases of the cylinder.

Formula Volume of a Cylinder

How to find the Volume of a Cylinder

This page examines the properties of a right circular cylinder . A cylinder has a radius (r) and a height (h) (see picture below).

Cylinder Volume Formula

Practice Problems on Area of a Cylinder

What is the volume of the cylinder with a radius of 2 and a height of 6?

Use the formula for the volume of a cylinder as shown below.

• Volume = Π *(r) 2 (h)
• Volume = Π *(2) 2 (6)

What is the volume of the cylinder with a radius of 3 and a height of 5?

• Volume = Π *(3) 2 (5)

What is the area of the cylinder with a radius of 6 and a height of 7?

Use the area of a cylinder formula .

• Volume = Π *(6) 2 (7)

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Cylinder is one of the basic 3d shapes, in geometry, which has two parallel circular bases at a distance. The two circular bases are joined by a curved surface, at a fixed distance from the center. The line segment joining the center of two circular bases is the axis of the cylinder. The distance between the two circular bases is called the height of the cylinder. LPG gas-cylinder is one of the real-life examples of cylinders. 

Since, the cylinder is a three-dimensional shape, therefore it has two major properties, i.e., surface area and volume. The total surface area of the cylinder is equal to the sum of its curved surface area and area of the two circular bases. The space occupied by a cylinder in three dimensions is called its volume.

Here we will learn about its definition, formulas, properties of cylinder and will solve some examples based on them. Apart from this figure, we have concepts of Sphere, Cone, Cuboid, Cube, etc. which we learn in Solid Geometry .

In mathematics, a cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance. These bases are normally circular in shape (like a circle ) and the center of the two bases are joined by a line segment, which is called the axis. The perpendicular distance between the bases is the height, “h” and the distance from the axis to the outer surface is the radius “r” of the cylinder.

Below is the figure of the cylinder showing area and height.

Cylinder Shape

A cylinder is a three-dimensional shape consisting of two parallel circular bases, joined by a curved surface. The center of the circular bases overlaps each other to form a right cylinder. The line segment joining the two centers is the axis, that denotes the height of the cylinder. 

The top view of the cylinder looks like a circle and the side view of the cylinder looks like a rectangle.

Unlike cones, cube and cuboid, a cylinder does not have any vertices, since the cylinder has a curved shape and no straight lines. It has two circular faces.

Each shape has some properties that differentiate one shape from another. Therefore, cylinders also have its characteristics. 

• The bases are always congruent and parallel.
• If the axis forms a right angle with the bases, which are exactly over each other, then it is called a “Right Cylinder”.
• It is similar to the prism since it has the same cross-section everywhere.
• If the bases are not exactly over each other but sideways, and the axis does not produce the right angle to the bases, then it is called “Oblique Cylinder”.
• If the bases are circular in shape, then it is called a right circular cylinder.
• If the bases are in an elliptical shape, then it is called an “Elliptical Cylinder”.

To learn more properties of cylinder .

The cylinder has three major quantities, based on which we have the formulas.

• Lateral Surface Area or curved surface area
• Total Surface Area

Let us see, their formulas now.

Curved Surface Area of Cylinder

The area of the curved surface of the cylinder which is contained between the two parallel circular bases. It is also stated as a lateral surface area. The formula for it is given by:

Total Surface Area of Cylinder

The total surface area of a cylinder is the sum of curved surafce area and the area of two circular bases. TSA = Curved surface + Area of Circular bases TSA = 2πrh + 2πr 2 We can see, from the above expression, 2πr is common. Therefore,

Volume of Cylinder

Every three dimensional shape or a solid has volume that occupies some space. The volume of the cylinder is the space occupied by it in any three-dimensional plane. The amount of water that could be immersed in a cylinder is described by its volume. The formula for the volume of cylinder is given by:

Where ‘h’ is the height and ‘r’ is the radius. Also, read:   Area Of Hollow Cylinder

Solved Examples

Question 1: Find the total surface area of the cylinder, whose radius is 5cm and height is 10cm?

Solution : We know, from the formula,

Total surface area of a cylinder, A = 2πr(r+h) square units

Therefore, A = 2π × 5(5 + 10) = 2π × 5(15) = 2π × 75 = 150 × 3.14 = 471 cm 2

Question 2:  What is the volume of a cylindrical shape water contain er, that has a height of 7cm and diameter of 10cm?

Solution: Given,

Diameter of the container = 10cm

Thus, the radius of the container = 10/2 = 5cm

Height of the container = 7cm

As we know, from the formula,

Volume of a cylinder = πr 2 h cubic units.

Therefore, volume of the given container, V = π × 5 2 × 7

V = π × 25 × 7 = (22/7) × 25 × 7 = 22 × 25

Frequently Asked Questions – FAQs

What are the characteristics of cylinder, how many vertices does a cylinder have, what are the real life examples of cylinder, what are the formulas of cylinder, how many bases does the cylinder have, how to find total surface area of cylinder.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

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Cylinder Volume Calculator

How to calculate volume of a cylinder, volume of a hollow cylinder, volume of an oblique cylinder.

Our cylinder volume calculator enables calculating the volume of that solid. Whether you want to figure out how much water fits in a can, coffee in your favorite mug, or even the volume of a drinking straw – you're in the right place. The other option is calculating the volume of a cylindrical shell (hollow cylinder).

Let's start from the beginning – what is a cylinder? It's a solid bounded by a cylindrical surface and two parallel planes. We can imagine it as a solid physical tin having lids on top and bottom. To calculate its volume, we need to know two parameters – the radius (or diameter) and height:

cylinder volume = π × cylinder radius² × cylinder height

The cylinder volume calculator helps in finding the volume of right, hollow and oblique cylinders:

The hollow cylinder, also called the cylindrical shell, is a three-dimensional region bounded by two right circular cylinders having the same axis and two parallel annular bases perpendicular to the cylinders' common axis.

It's easier to understand that definition by imagining, e.g., a drinking straw or a pipe – the hollow cylinder is this plastic, metal, or other material. The formula behind the volume of a hollow cylinder is:

cylinder_volume = π × (R² - r²) × cylinder_height

where R – external radius, and r – internal radius

Similarly, we can calculate the cylinder volume using the external diameter, D , and internal diameter, d , of a hollow cylinder with this formula:

cylinder_volume = π × [(D² - d²)/4] × cylinder_height

To calculate the volume of a cylindrical shell, let's take some real-life examples, maybe... a roll of toilet paper, because why not? 😀

Enter the external diameter of the cylinder . The standard is equal to approximately 11 cm.

Determine the internal cylinder diameter . It's the internal diameter of the cardboard part, around 4 cm.

Find out what's the height of the cylinder ; for us, it's 9 cm.

Tadaaam! The volume of a hollow cylinder is equal to 742.2 cm³.

Remember that the result is the volume of the paper and the cardboard. If you want to calculate how much plasticine you can put inside the cardboard roll, use the standard formula for the volume of a cylinder – the calculator will calculate it in the blink of an eye!

The oblique cylinder is the one that 'leans over' – the sides are not perpendicular to the bases in contrast to a standard 'right cylinder'. How to calculate the volume of an oblique cylinder? The formula is the same as for the straight one. Just remember that the height must be perpendicular to the bases.

Now that you know how to calculate a volume of a cylinder, maybe you want to determine the volumes of other 3D solids? Use this general volume calculator !

If you are curious about how many teaspoons or cups fit into your container, use our volume converter .

To calculate the volume of soil needed for flower pots of different shapes – also for the cylindrical one – use the potting soil calculator .

Where can you find cylinders in nature?

Cylinders are all around us , and we are not just talking about Pringles’ cans. Although things in nature are rarely perfect cylinders, some examples are tree trunks & plant stems, some bones (and therefore bodies), and the flagella of microscopic organisms. These make up a large amount of the natural objects on Earth!

How do you draw a cylinder?

To draw a cylinder, follow these steps:

Draw a slightly flattened circle. The more flattened it is, the closer you are to looking at the cylinder side on .

Draw two equal, parallel lines from the far sides of your circle going down.

Link the ends of the two lines with a semi-circular line that looks the same as the bottom half of your top circle.

Add shadow and shading as appropriate.

How do you calculate the weight of a cylinder?

To calculate the weight of a cylinder:

Square the radius of the cylinder .

Multiply the square of radius by pi and the cylinder’s height .

Multiply the volume by the density of the cylinder. The result is the cylinder’s weight.

How do you calculate the surface area to volume ratio of a cylinder?

Find the volume of the cylinder using the formula πr²h .

Find the surface area of the cylinder using the formula 2πrh + 2πr² .

Make a ratio out of the two formulas, i.e., πr²h : 2πrh + 2πr² .

Alternatively, simplify it to rh : 2(h+r) .

Divide both sides by one of the sides to get the ratio in its simplest form.

How do you find the height of a cylinder?

If you have the volume and radius of the cylinder:

• Make sure the volume and radius are in the same units (e.g., cm³ and cm).

Square the radius.

• Divide the volume by the radius squared and pi to get the height in the same units as the radius.

If you have the surface area and radius (r):

• Make sure the surface and radius are in the same units .
• Subtract 2πr² from the surface area.
• Divide the result of step 2 by 2πr.
• The result is the height of the cylinder.

How do I find the radius of a cylinder?

If you have the volume and height of the cylinder:

• Make sure the volume and height are in the same units (e.g., cm³ and cm).
• Divide the volume by pi and the height.
• Square root the result.

If you have the surface area and height (h):

• Substitute the height, h, and surface area into the equation, surface area = 2πrh + 2πr².
• Divide both sides by 2π.
• Subtract surface area/2π from both sides.
• Solve the resulting quadratic equation.
• The positive root is the radius.

How do you find the volume of an oval cylinder?

To find the volume of an oval cylinder:

Multiply the smallest radius of the oval (minor axis) by its largest radius (major axis).

Multiply this new number by pi .

Divide the result of step 2 by 4. The result is the area of the oval.

Multiply the area of the oval by the height of the cylinder.

The result is the volume of an oval cylinder.

How do you find the volume of a slanted cylinder?

To calculator the volume of a slanted cylinder:

Find the radius, side length, and slant angle of the cylinder.

Multiply the result by pi.

Take the sin of the angle .

Multiply the sin by the side length.

Multiply the result from steps 3 and 5 together.

The result is the slanted volume.

How do you calculate the swept volume of a cylinder?

To compute the swept volume of a cylinder:

Divide the bore diameter by 2 to get the bore radius .

Square the bore radius.

Multiply the square radius by pi.

Multiply the result of step 3 by the length of the stroke . Make sure the units for bore and stroke length are the same.

The result is the swept volume of one cylinder.

Car vs. Bike

Exponential function.

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VOLUME OF CYLINDER WORD PROBLEMS WORKSHEET

1. The cylindrical Giant Ocean Tank at the New England Aquarium in Boston is 24 feet deep and has a radius of 18.8 feet. Find the volume of the tank. Use the approximate of value of  π , that is 3.14 and round your answer to the nearest tenth if necessary.

2. A standard-size bass drum has a diameter of 22 inches  and is 18 inches deep. Find the volume of this drum. Use the approximate of value of    ∏ , that is 3.14 and round your answer to the nearest tenth if necessary.

3. A barrel of crude oil contains about 5.61 cubic feet of oil. How many barrels of oil are contained in 1 mile of a pipeline that has an inside diameter of 6 inches and is completely filled with oil ? How much is “1 mile” of oil in this pipeline worth at a price of $100 per barrel ?

4. A pan for baking French bread is shaped like half a cylinder as shown in the figure. Find the volume of uncooked dough that would fill this pan. Use the approximate of value of    ∏ , that is 3.14 and round your answer to the nearest tenth if necessary.

1. Answer :

Step 1 : 

Because the tank is in the shape of cylinder, we can use the formula of volume of a cylinder to find volume of the tank.

V  =   π r 2 h cubic units

Step 2 : 

Substitute  the given measures. 

V   ≈  3.14  · 18.8 2 · 24

(Here deep 24 feet is considered as height)

V  ≈  3.14 · 353.44 · 24

V  ≈  26635.2

So, the volume of the tank is about 26635.2 cubic feet.

2. Answer :

Usually the bass drum would be in the shape of cylinder. So, we can use the formula of volume of a cylinder, to find volume of the bass drum.

V  =   π r 2 h cubic units  -----(1)

To find the volume, we need the radius of the cylinder. But, the diameter is given, that is 22 in. So, find the radius. 

r  =  diameter / 2

r  =  22/2

r  =  11

Substitute  π    ≈  3.14,  r = 11 and h = 18  in (1) .

V  ≈  3.14  · 11 2  · 18

(Here deep 18 inches is considered as height)

V  ≈  3.14 · 121 · 18

V  ≈  6838.9

So, the volume of the bass drum is about 6838.9 cubic inches.

3. Answer :

Usually the pipe line would be in the shape of cylinder. So, we can use the formula of volume of a cylinder to find volume of the crude oil in the pipe line. 

To find the volume, we need the radius of the cylinder. But, the diameter is given, that is 6 in. So, find the radius. 

r  =  6/2

r  =  3 inches

Convert the inches into feet by multiplying 1/12.

Because, 

1 inch  =  1/12 feet

So, we have 

r  =  3 x 1/12 feet

r  =  1/4 feet

Convert the length of the pipeline from miles to feet.

1 mile  =  5280 feet

length  =  1 mile

length  =  1 x 5280 feet

length  =  5280 feet

Substitute  π    ≈  3.14,  r = 1/4 and h = 5280   in (1).

V   ≈  3.14  · (1/4) 2  · 5280

(Here , the length 5280 feet is considered as height)

V  ≈  3.14 · (1/16) · 5280

V  ≈  1036.2 cubic feet

To find how many  barrels of oil are contained in 1 mile of a pipeline, divide the volume of crude oil in the pipeline (1036.2 cu.ft) by 5.61.

Because a barrel of crude oil contains about 5.61 cubic feet of oil.

So, n umber of  barrels of oil are contained in 1 mile of a pipeline is 

=  1036.2 / 5.61

=  184.7

There are about 184.7  barrels of oil are contained in 1 mile of a pipeline.

Find the worth of  “1 mile” of oil in the pipeline at a price of $100 per barrel.

No. of barrels of oil in 1 mile of a pipeline  =  184.7  

So, the worth of  “1 mile” of oil in the pipeline is 

=  $100 x 184.7

=  $18,470

The worth of “1 mile” of oil in the pipeline at a price of $100 per barrel is about $18,470.

4. Answer :

Solution : 

Because the pan is shaped like half a cylinder, we  can use the formula of volume of a cylinder to find volume of  uncooked dough that would fill this pan

(Because the pan is shaped like half a cylinder,  1/2 is multiplied by the formula of volume of a cylinder)

To find the volume, we need the radius of the cylinder. But, the diameter is given, that is 3.5 in. So, find the radius. 

r  =  3.5/2

r  =  1.75

Substitute  π   ≈ 3.14,  r = 1.75 and h = 12  in (1) .

V  ≈   1/2   ·  3.14  · 1.75 2 · 12

V   ≈   1/2   ·  3.14  · 3.0625 · 12

V   ≈  57.7

So,  the volume of uncooked dough that would fill the pan is about 57.7 cu.inches.

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GMAT Math : Cylinders

Study concepts, example questions & explanations for gmat math, all gmat math resources, example questions, example question #1 : calculating the surface area of a cylinder.

What is the surface area of a cylinder with a radius of 7 and a height of 3?

All we really need here is to remember the formula for the surface area of a cylinder.

Example Question #2 : Calculating The Surface Area Of A Cylinder

The height of a cylinder is twice the circumference of its base. The radius of the base is 9 inches. What is the surface area of the cylinder?

Calculate the surface area of the following cylinder.

The equation for the surface area of a cylinder is:

Calculate the surface area of the following cylinder.

The equation for the surface area of a cylinder is 

Note:  we were given the diameter of the cylinder (10), in order to find the radius we had to divide the diameter by two. 

Example Question #5 : Calculating The Surface Area Of A Cylinder

A cylinder has a height of 9 and a radius of 4. What is the total surface area of the cylinder?

We are given the height and the radius of the cylinder, which is all we need to calculate its surface area. The total surface area will be the area of the two circles on the bottom and top of the cylinder, added to the surface area of the shaft. If we imagine unfolding the shaft of the cylinder, we can see we will have a rectangle whose height is the same as that of the cylinder and whose width is the circumference of the cylinder. This means our formula for the total surface area of the cylinder will be the following:

Example Question #6 : Calculating The Surface Area Of A Cylinder

This question is looking for the surface area of a cylinder with only 1 base. Our surface area of a cylinder is given by:

Example Question #7 : Calculating The Surface Area Of A Cylinder

To find the surface area of a cylinder, you must use the following equation.

Example Question #8 : Calculating The Surface Area Of A Cylinder

The surface area of a cylinder can be calculated from its radius and height as follows:

Example Question #1 : Calculating The Volume Of A Cylinder

What is the volume of a cone with a radius of 6 and a height of 7?

The only tricky part here is remembering the formula for the volume of a cone. If you don't remember the formula for the volume of a cone, you can derive it from the volume of a cylinder. The volume of a cone is simply 1/3 the volume of the cylinder. Then,

What is the volume of a sphere with a radius of 9?

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Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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