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Volume of a sphere

# Volume of a sphere

Here you will learn about the volume of a sphere, including how to calculate the volume of a sphere given its radius and how to find the volume of a hemi-sphere.

Students will first learn about the volume of a sphere as a part of geometry in 8 th grade and will expand on their understanding in high school.

## What is the volume of a sphere?

The volume of a sphere is the amount of space inside a sphere. A sphere is a three-dimensional shape that is perfectly round, with no edges or vertices.

The radius of a sphere is the fixed distance from the center of the sphere to any given point on its surface. The diameter of the sphere is a straight line that passes through the center and connects two points on the sphere.

To calculate the volume of a sphere, use the formula:

V=\cfrac{4}{3} \; \pi r^3

Notice the cube of the radius \mathrm(r^3) in the volume formula. Volume is a measure in three-dimensions so the units for the volume are cubic units. For example, cubic feet \mathrm(ft^3), cubic inches \mathrm(in^3) and cubic centimeters \mathrm(cm^3).

For example,

Find the volume of the sphere with a radius of 5~{cm}.

\begin{aligned} \text{Volume}&=\frac{4}{3} \pi r^3 \\\\ &= \frac{4}{3} \times \pi \times 5^3\\\\ &=\frac{500}{3}\pi\\\\ &=523.6 \ cm^3 \ \text{(to 1 dp)}\\\\ \end{aligned}

The answer should be stated in cubic centimeters because the units used are centimeters.

### Calculating the radius given the volume

In order to calculate the radius of a sphere given the volume, you need to rearrange the formula for the volume of a sphere (V=\cfrac{4}{3} \; \pi r^3) to make r (the radius) the subject of the formula.

For example,

Now you have the formula for determining the radius of a sphere given the volume as:

r=\sqrt[3]{\cfrac{3V}{4\pi}}

## Common Core State Standards

How does this relate to 8 th grade and high school math?

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

• High school: Geometry (HS.G.GMD.A.3)
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

## How to calculate the volume of a sphere

In order to calculate the volume of a sphere:

1. Write down the formula for the volume of a sphere.
2. Substitute given values into the formula.
3. Complete the calculation.
4. Write the answer, including the units.

## Volume of a sphere examples

### Example 1: volume of a sphere given the radius

1. Write down the formula for the volume of a sphere.

The formula for the volume of a sphere is V=\cfrac{4}{3} \; \pi r^3.

2Substitute given values into the formula.

Substitute the value of the radius r=6 into the formula.

V=\cfrac{4}{3} \times \pi \times 6^3

3Complete the calculation.

Use a calculator to find the volume.

V=288 \pi =904.7786842…

4Write the answer, including the units.

Here you are asked to give the answer to 1 decimal place.

The volume of the sphere is: 904.8 \mathrm{~cm}^3 \; (1dp).

Write down the formula for the volume of a sphere.

Substitute given values into the formula.

Complete the calculation.

Write the answer, including the units.

### Example 3: volume of a sphere given the diameter – in terms of pi (π)

Calculate the volume of a sphere with a diameter of 6~{m}. Write your answer in terms of \pi.

Write down the formula for the volume of a sphere.

Substitute given values into the formula.

Complete the calculation.

Write the answer, including the units.

### Example 4: volume of a hemisphere

Write down the formula for the volume of a sphere.

Substitute given values into the formula.

Complete the calculation.

Write the answer, including the units.

### Example 5: volume of a hemisphere

Find the volume of a hemisphere with radius 12.5 \mathrm{~mm}.

Write down the formula for the volume of a sphere.

Substitute given values into the formula.

Complete the calculation.

Write the answer, including the units.

## How to calculate the radius of a sphere given the volume

In order to calculate the radius of the sphere given the volume:

1. Write down the formula for the radius of a sphere, in terms of the volume.
2. Substitute the given values into the formula.
3. Complete the calculation.
4. Write the answer, including the units.

## Calculating the radius of a sphere examples

### Example 6: calculating the radius given the volume

The total volume of a sphere is 3,000 \mathrm{~cm}^3. Calculate the radius of the sphere, correct to 2 decimal places.

Write down the formula for the radius of a sphere, in terms of the volume.

Substitute the given values into the formula.

Complete the calculation.

Write the answer, including the units.

### Example 7: calculating the radius given the volume

Calculate the radius of a sphere with the volume 8,460 m^3. Write your answer to the nearest centimeter.

Write down the formula for the radius of a sphere, in terms of the volume.

Substitute the given values into the formula.

Complete the calculation.

Write the answer, including the units.

### Teaching tips for volume of a sphere

• Make sure to use visual representations, such as diagrams and physical spheres, to allow students to recognize the shape and structure of spheres.

• Work through examples, giving students step-by-step instructions on how to find the volume of the sphere. Displaying it within the classroom will give students a place to refer back to when needed.

• While worksheets with practice questions have a place within the classroom, allow students to use interactive tools and engage in conversations with other students about different types of problems.

### Easy mistakes to make

• Using the incorrect formula
Students are learning several formulas during middle school and high school geometry that can be used. It’s important that students have a general basis of each formula, so they know which to use in each mathematical situation.

• Rounding too soon during the calculation
It is important to not round the answer until the end of the calculation. This will mean your final answer is accurate. It is useful to keep your answer in terms of \pi until you round the answer at the very end of the question.

• Using the wrong units
For area, use square units such as \mathrm{cm}^2. For volume, use cube units such as \mathrm{cm}^3. An easy way for students to remember this is that you find the area of 2 D shapes and volume of 3 D shapes.

• Mixing up the radius and the diameter
It is a common error to mix up radius and diameter. Remember the radius is half of the diameter, and the diameter stretches across the entire sphere.

### Practice volume of a sphere questions

1. Calculate the volume of the sphere. Write your answer to 1 decimal place.

4188.8 \mathrm{~cm}^3

314.2 \mathrm{~cm}^3

1.3 \mathrm{~cm}^3

1256.6 \mathrm{~cm}^3

You are finding the volume of a sphere so substitute the value of r into the formula.

\begin{aligned} V&=\cfrac{4}{3} \pi r^3 \\\\ &= \cfrac{4}{3} \times \pi \times 10^3\\\\ &=\cfrac{4000}{3}\pi\\\\ &=4188.790205… \\\\ &=4188.8 \ cm^3 \ \text{(1dp)}\\\\ \end{aligned}

2. Calculate the volume of the sphere with the diameter 13.6~{cm}.

10,536.7 \mathrm{~cm}^3

2,324.3 \mathrm{~cm}^3

1,317.1 \mathrm{~cm}^3

581.1 \mathrm{~cm}^3

The radius is half the value of the diameter, and so,

r=13.6 \div 2=6.8 \mathrm{~cm}

Substituting r=6.8 into V=\cfrac{4}{3} \pi r^3, we have

\begin{aligned} V&=\cfrac{4}{3} \pi r^3 \\\\ &= \cfrac{4}{3} \times \pi \times 6.8^3\\\\ &=1317.089682… \\\\ &=1317.1 \; \text{cm}^3\text{ (1dp)} \end{aligned}

3. A sphere has a radius of 9~{m}. Calculate the volume of the sphere in terms of \pi.

243\pi \mathrm{~cm^3}

108\pi \mathrm{~cm^3}

972\pi \mathrm{~cm^3}

729\pi \mathrm{~cm^3}

Substituting r=9 into the formula for the volume of a sphere, you have:

\begin{aligned} V&=\cfrac{4}{3} \pi r^3 \\\\ &= \cfrac{4}{3} \times \pi \times 9^3\\\\ &=972\pi \\\\ \end{aligned}

4. Calculate the volume of the hemisphere. Write your answer to the nearest centimeter.

33,510 \mathrm{~mm}^3

16,755 \mathrm{~mm}^3

1,257 \mathrm{~mm}^3

3,770 \mathrm{~mm}^3

When finding the volume of a hemisphere, calculate half of the volume of a sphere.

\begin{aligned} V&=(\cfrac{4}{3} \pi r^3) \div{2}\\\\ &= (\cfrac{4}{3} \times \pi \times 20^3) \div{2}\\\\ &=\cfrac{16 000}{3} \pi\\\\ &=16 755.16082… \\\\ &=16 755{~mm}^3 (0dp) \end{aligned}

5. A hemisphere has a diameter of 24 \mathrm{~km}. Calculate the volume of the hemisphere in terms of \pi.

2304 \pi \mathrm{~km^3}

7240 \mathrm{~km^3}

1152\pi \mathrm{~km^3}

3620 \mathrm{~km^3}

When finding the volume of a hemisphere, calculate half of the volume of a sphere.

\begin{aligned} V&=\cfrac{4}{3} \pi r^3 \div{2}\\\\ &= \cfrac{4}{3} \times \pi \times 12^3 \div{2}\\\\ &=1152\pi \\\\ &=1152\pi \ km^3\\\\ \end{aligned}

6. A sphere has a volume of 1500 \mathrm{~cm}^3. Calculate the radius of the sphere, correct to 1 decimal place.

7.1 \mathrm{~cm}

18.9 \mathrm{~cm}

37.8 \mathrm{~cm}

11.3 \mathrm{~cm}

Using the formula for the radius in terms of the volume, substitute the value of the volume and solve to find the radius.

\begin{aligned} & r=\sqrt[3]{\cfrac{3 V}{4 \pi}} \\\\ & =\sqrt[3]{\cfrac{3 \times 1500}{4 \pi}}\\\\ & =\sqrt[3]{358.098622 \ldots} \\\\ & =7.101240423 \ldots \\\\ & =7.1 \mathrm{~cm} \; (1 \mathrm{dp}) \end{aligned}

## Volume of a sphere FAQs

What is a sphere?

A sphere is a set of points in space that are equidistant distance (r) from the center. It is commonly known as a 3 D solid shape that has no sides or vertices.

What is the formula for the volume of a sphere?

The formula for the volume of a sphere is V=\cfrac{4}{3} \pi r^3, where r is the radius of the sphere.

How do you find the surface area of a sphere?

To find the surface area of a sphere, you will use the formula A=4 \pi r^2, where r is the radius and \pi is the mathematical constant.

What is the Archimedes’ principle?

The Archimedes’ principle is a fundamental principle of physics, discovered by the ancient Greek mathematician, Archimedes, that states when a body is partially or fully submerged in a fluid, it experiences an upward buoyant force that is equal to the weight of the fluid displaced by the body.

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