Math resources Geometry Volume

Volume of a hemisphere

# Volume of a hemisphere

Here you will learn about the volume of a hemisphere, including what a hemisphere is and how to calculate the volume of a hemisphere given the radius or the diameter.

Students will first learn about the volume of a hemisphere as part of geometry in 8 th grade, then expand their learning in high school.

## What is the volume of a hemisphere?

The volume of a hemisphere is the amount of space inside a hemisphere.

To calculate the volume of a hemisphere, you need to know what a hemisphere is. “Hemi” means half, so a hemisphere is half of a sphere. It is a three dimensional shape, with a radius, r.

To calculate the volume of a full sphere, where r is the radius of the sphere, you use the formula

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

Since a hemisphere is half of a sphere, to find the volume of a hemisphere you halve the volume of a sphere.

Here is the volume of a hemisphere formula, with radius r.

V=\cfrac{4}{3} \, \pi{r^3}\div{2}

Or, alternatively,

\begin{aligned}\text {Volume of a hemisphere}&=\cfrac{1}{2}\times\cfrac{4}{3} \, \pi{r^3} \\\\ &=\cfrac{2}{3} \, \pi{r^3} \end{aligned}

## Common Core State Standards

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

• Grade 8 – Geometry (8.G.C.9)
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 hemisphere

In order to calculate the volume of a hemisphere:

1. Write down the formula for the volume of a hemisphere.
2. Substitute in the known variables.
3. Solve for the unknown variable.

## Volume of a hemisphere examples

### Example 1: radius is given

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

The formula for finding the volume of a hemisphere is

V=\cfrac{4}{3} \, \pi{r^3}\div{2}

2Substitute in the known variables.

V=\cfrac{4}{3} \, \pi (4.5)^3\div 2

3Solve for the unknown variable.

Using a calculator, V=190.9\mathrm{~cm}^3 (nearest tenth)

### Example 2: diameter of a hemisphere is given

Calculate the volume of a hemisphere with diameter 12.3\mathrm{~mm}. Give your answer to the nearest tenth.

Write down the formula for the volume of a hemisphere.

Substitute in the known variables.

Solve for the unknown variable.

Write down the formula for the volume of a hemisphere.

Substitute in the known variables.

Solve for the unknown variable.

### Example 4: leaving answer in terms of π, diameter of a hemisphere is given

Calculate the volume of a hemisphere with diameter 6\mathrm{~m}. Leave your answer in terms of \pi.

Write down the formula for the volume of a hemisphere.

Substitute in the known variables.

Solve for the unknown variable.

### Example 5: finding the radius, given the volume

The volume of a hemisphere is 500\mathrm{~cm}^3. Calculate the radius. Give your answer to the nearest hundredth.

Write down the formula for the volume of a hemisphere.

Substitute in the known variables.

Solve for the unknown variable.

### Example 6: finding the diameter, given the volume

The volume of a hemisphere is 400\mathrm{~cm}^3. Calculate the diameter. Give your answer to the nearest tenth.

Write down the formula for the volume of a hemisphere.

Substitute in the known variables.

Solve for the unknown variable.

### Example 7: finding the capacity of a hemisphere

A basin is filled with water. The basin is a hemisphere with a radius of 30\mathrm{~cm}. Find the capacity of the hemisphere in liters.

Write down the formula for the volume of a hemisphere.

Substitute in the known variables.

Solve for the unknown variable.

### Teaching tips for volume of a hemisphere

• Start by introducing real life examples of hemispheres, such as a dome, igloo, or even the northern and southern hemispheres of earth. This helps students visualize the shape and understand its relevance.

• Use visual aids such as diagrams and physical models to help students visualize a hemisphere and its volume. Showing them how slicing a sphere into two equal halves can make the idea more concrete.

• Provide worksheets with a variety of practice problems at different difficulty levels. Include problems that involve finding the volume of a hemisphere given the radius, as well as word problems and real life applications.

### Easy mistakes to make

• Using the incorrect formula
There are several formulas relating to circles that can be confused so make sure you learn the difference between them.

• Rounding the answer too early
It is important to wait to round the answer until the end of the calculation. It is useful to keep your answer in terms of \pi until you round the answer at the very end of the question.

• Writing the incorrect units
For area, you use square units such as square centimeters. You can write this with an exponent as cm^2.
For volume, you use cubic units such as cubic centimeters. You can write this with an exponent as cm^3.

• Mixing up the radius and diameter
It is a common error to mix up radius and diameter. Remember the radius is half of the diameter.

### Practice volume of a hemisphere questions

1. Find the volume of a hemisphere with radius 7.9\mathrm{~cm}. Give your answer correct to the nearest tenth.

1032.6\mathrm{~cm}^3

2065.2\mathrm{~cm}^3

2065.3\mathrm{~cm}^3

1032.7\mathrm{~cm}^3

The volume of the hemisphere can be found by using the volume of a sphere and halving it.

\begin{aligned}V&=\cfrac{4}{3} \, \pi{r^3}\div{2} \\\\ V&=\cfrac{4}{3} \, \pi(7.9)^3\div{2} \\\\ V&=1032.618… \\\\ V&=1032.6\mathrm{~cm}^3\text{ (nearest tenth)} \end{aligned}

2. Find the volume of a hemisphere with diameter 4.6\mathrm{~m}. Give your answer correct to the nearest tenth.

25.4\mathrm{~m}^3

203.8\mathrm{~m}^3

203.9\mathrm{~m}^3

25.5\mathrm{~m}^3

First you need to divide the diameter by 2 to find the radius. The radius is 2.3\mathrm{~m}. The volume of the hemisphere can be found by using the volume of a sphere and halving it.

\begin{aligned}V&=\cfrac{2}{3} \, \pi{r^3} \\\\ V&=\cfrac{2}{3} \, \pi(2.3)^3 \\\\ V&=25.4825… \\\\ V&=25.5\mathrm{~m}^3\text{ (nearest tenth)} \end{aligned}

3. Find the volume of a hemisphere with radius 10\mathrm{~mm}. Leave your answer in terms of \pi.

\cfrac{2000}{3} \, \pi\mathrm{~mm}^3

1000 \, \pi\mathrm{~mm}^3

\cfrac{1000}{3} \, \pi\mathrm{~mm}^3

2000 \, \pi\mathrm{~mm}^3

The volume of the hemisphere can be found by using the volume of a sphere and halving it.

\begin{aligned}V&=\cfrac{4}{3} \, \pi{r^3}\div{2} \\\\ V&=\cfrac{4}{3} \, \pi(10)^3\div{2} \\\\ V&=\cfrac{2000}{3} \, \pi \\\\ V&=\cfrac{2000}{3} \, \pi\mathrm{~mm}^3 \end{aligned}

4. Find the volume of a hemisphere with diameter 4\mathrm{~km}. Leave your answer in terms of \pi.

\cfrac{256}{3} \, \pi\mathrm{~km}^3

\cfrac{16}{3} \, \pi\mathrm{~km}^3

\cfrac{32}{3} \, \pi\mathrm{~km}^3

\cfrac{128}{3} \, \pi\mathrm{~km}^3

First you need to divide the diameter by 2 to find the radius. The radius is 2 \, km. The volume of the hemisphere can be found by using the formula:

\begin{aligned}V&=\cfrac{2}{3} \, \pi{r^3} \\\\ V&=\cfrac{2}{3} \, \pi(2)^3 \\\\ V&=\cfrac{16}{3} \, \pi \\\\ V&=\cfrac{16}{3} \, \pi\mathrm{~km}^3 \end{aligned}

5. A hemisphere has a volume of 1500\mathrm{~cm}^3. Calculate its radius. Give your answer correct to the nearest hundredth.

17.84\mathrm{~cm}

17.85\mathrm{~cm}

8.94\mathrm{~cm}

8.95\mathrm{~cm}

Equate the volume of a hemisphere to the given volume.

\begin{aligned}1500&=\cfrac{2}{3} \, \pi{r^3} \\\\ 4500&=2 \, \pi{r^3} \\\\ 2250&=\pi{r^3} \\\\ \cfrac{2250}{\pi}&=r^{3} \\\\ r&=\sqrt[3]{\cfrac{2250}{\pi}} \\\\ r&=8.94700… \\\\ r&=8.95\mathrm{~cm}\text{ (nearest hundredth)} \end{aligned}

6. A hemisphere has a volume of 2300\mathrm{~cm}^3. Calculate its diameter. Give your answer correct to the nearest tenth.

10.3\mathrm{~cm}

20.6\mathrm{~cm}

10.4\mathrm{~cm}

20.7\mathrm{~cm}

Form an equation by substituting the volume into the formula for the volume of a hemisphere and solve it to find the radius. Finally, you multiply the radius by 2 to find the diameter.

\begin{aligned}2300&=\cfrac{4}{3} \, \pi{r^3}\div{2} \\\\ 4600&=\cfrac{4}{3} \, \pi{r^3} \\\\ 13800&=4 \, \pi{r^3} \\\\ 3450&=\pi{r^3} \\\\ \cfrac{3450}{\pi}&=r^3 \\\\ r&=\sqrt[3]{\cfrac{3450}{\pi}} \\\\ r&=10.317… \end{aligned}

\begin{aligned} d&=r\times{2}\\\\ &=10.317…\times{2}\\\\ &=20.6\mathrm{~cm}\text{ (nearest tenth)} \end{aligned}

7. The capacity of a garden water feature is 20\mathrm{~L}. The water feature is a hemisphere. Determine the radius of the water feature to the nearest hundredth.

20.22\mathrm{~cm}

21.22\mathrm{~cm}

24.75\mathrm{~cm}

21.01\mathrm{~cm}

Convert liters to cubic centimeters first: 20\mathrm{~L}=20~000\mathrm{~cm}^3

\begin{aligned}V&=\cfrac{2}{3} \, \pi{r^3} \\\\ 20~000&=\cfrac{2}{3} \, \pi\times{r^3} \\\\ 60~000&=2 \, \pi\times{r^3} \\\\ 30~000&=\pi\times{r^3} \\\\ \cfrac{30~000}{\pi}&=r^{3} \\\\ \sqrt[3]{\cfrac{30~000}{\pi}}&=r \\\\ r&=21.22\mathrm{~cm}\text{ (nearest hundredth)} \end{aligned}

## Volume of a hemisphere FAQs

What is a hemisphere shape?

A hemisphere is a 3D shape that looks like half of a sphere. It has a flat circular base and a curved surface extending from the base in all directions.

How do you find the volume of a hemisphere?

To find the volume of a hemisphere, you can use the formula for half of the volume of a sphere: V=\cfrac{2}{3} \, \pi{r^3}

What is the difference between the volume of a hemisphere and the curved surface area of a hemisphere?

The volume of a hemisphere measures the amount of space enclosed by its curved surface and base, calculated using the formula V=\cfrac{2}{3} \, \pi{r^3}.

The curved surface area of a hemisphere refers only to the area of its curved surface, excluding the base. It is calculated using the formula A=2 \, \pi{r^2}.

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