Volume of a Hemisphere: Formula, Examples, and How to Find It

The volume of a hemisphere is the amount of space inside half of a sphere. A hemisphere is created when a sphere is cut into two equal parts through its center.

The volume of a hemisphere formula is V = 2/3πr³, where r is the radius and π is approximately 3.14. Because a hemisphere is exactly half of a sphere, its volume is also half of the sphere’s volume.

In this guide, you will learn how to find the volume of a hemisphere, how to use radius and diameter, solve step-by-step examples, and avoid common mistakes involving units and formulas.

How to Find the Volume of a Hemisphere

To find the volume of a hemisphere, you need to know its radius. Then use:

V = 2/3πr³

The formula comes directly from the volume of a sphere:

Sphere: V = 4/3πr³

A hemisphere is half of a sphere, so:

Hemisphere: V = 1/2 × 4/3πr³ = 2/3πr³

This means you can find the volume by cubing the radius, multiplying by π and 2, and then dividing by 3.

Try the interactive hemisphere explorer below. Move the slider to change the radius and see how the hemisphere’s volume changes.

The interactive model makes one important idea easy to see: as the radius increases, the volume increases much faster, because the radius is cubed in the formula.

For example, doubling the radius does not simply double the volume. It makes the volume 8 times larger, because:

(2r)³ = 8r³

What Is the Volume of a Hemisphere?

The volume of a hemisphere is the amount of three-dimensional space contained inside a hemisphere.

A hemisphere is exactly half of a sphere. The word “hemi” means half. If you cut a sphere through its center, you create two equal hemispheres.

A hemisphere has:

• One curved surface
• One flat circular base
• A radius extending from the center of the original sphere to the curved surface
• A diameter across the circular base

The volume measures the space inside the shape. It is different from the surface area, which measures the outside surfaces.

Hemisphere vs. Sphere

The easiest way to understand hemisphere volume is to compare it with sphere volume.

If two identical hemispheres are put together along their flat circular bases, they form one complete sphere. Therefore, two identical hemispheres have the same total volume as one sphere.

Volume vs. Surface Area

Do not confuse the volume of a hemisphere with its surface area.

The formulas are different because volume measures three-dimensional space, while surface area measures two-dimensional surfaces.

For volume, the radius is cubed. Therefore, the answer must be written in cubic units, such as cm³, m³, or in³.

Volume of a Hemisphere Formula

The standard volume of a hemisphere formula is:

V = 2/3πr³

Where:

• V = volume of the hemisphere
• π = pi, approximately 3.14
• r = radius of the hemisphere

Why Is the Formula 2/3πr³?

Start with the formula for a sphere:

V = 4/3πr³

A hemisphere is exactly half of a sphere, so divide the sphere’s volume by 2:

V = 1/2 × 4/3πr³

Therefore:

V = 2/3πr³

This is why the coefficient in the hemisphere formula is 2/3, rather than 4/3.

What If the Diameter Is Given?

Sometimes a problem gives you the diameter instead of the radius.

Remember:

d = 2r

Therefore:

r = d/2

If the diameter is 12 cm, the radius is:

r = 12 ÷ 2 = 6 cm

Then use the standard formula:

V = 2/3π(6)³

You should always convert diameter to radius before using V = 2/3πr³.

You can also write the hemisphere volume directly in terms of diameter:

V = 1/12πd³

However, for most students, converting the diameter to radius first is easier and helps avoid mistakes.

How to Find the Volume of a Hemisphere

You can find the volume of a hemisphere in three simple steps.

Step 1: Find the Radius

Look at the information given in the problem.

If the radius is given, use it directly.

If the diameter is given, divide it by 2:

r = d ÷ 2

For example, if:

d = 14 cm

then:

r = 14 ÷ 2 = 7 cm

Step 2: Use the Hemisphere Volume Formula

Write:

V = 2/3πr³

Then substitute the radius.

For r = 7 cm:

V = 2/3π(7)³

Step 3: Calculate the Volume

First cube the radius:

7³ = 343

Then:

V = 2/3 × π × 343

Using π ≈ 3.14:

V ≈ 718.38 cm³

So the volume of the hemisphere is approximately:

718.38 cm³

Always use cubic units for volume.

Quick Formula Guide

Volume of a Hemisphere: Worked Examples

Example 1: Find the Volume From the Radius

Problem: Find the volume of a hemisphere with a radius of 5 cm. Use π ≈ 3.14.

Step 1: Write the formula.

V = 2/3πr³

Step 2: Substitute r = 5.

V = 2/3 × 3.14 × 5³

Step 3: Cube the radius.

5³ = 125

So:

V = 2/3 × 3.14 × 125

V ≈ 261.67 cm³

Answer: 261.67 cm³

The volume of the hemisphere is approximately 261.67 cubic centimeters.

Example 2: Find the Volume From the Diameter

Problem: A hemisphere has a diameter of 12 cm. What is its volume?

The formula requires the radius, so first find the radius.

r = d ÷ 2

r = 12 ÷ 2 = 6 cm

Now use:

V = 2/3πr³

Substitute:

V = 2/3 × 3.14 × 6³

Since:

6³ = 216

we get:

V = 2/3 × 3.14 × 216

V ≈ 452.16 cm³

Answer: 452.16 cm³

This is a common type of hemisphere problem. The most important step is remembering to divide the diameter by 2 before using the radius-based formula.

Example 3: Compare a Sphere and a Hemisphere

Problem: A sphere and a hemisphere have the same radius of 6 inches. How much larger is the sphere’s volume?

Sphere:

V = 4/3π(6)³

Hemisphere:

V = 2/3π(6)³

Because the hemisphere is exactly half of the sphere:

Sphere volume = 2 × hemisphere volume

Using π ≈ 3.14:

Sphere:

V ≈ 904.32 in³

Hemisphere:

V ≈ 452.16 in³

Therefore, the sphere has twice the volume of the hemisphere.

Example 4: A Real-World Hemisphere

Imagine a bowl shaped like a hemisphere with a radius of 10 cm. How much space can it hold?

Use:

V = 2/3πr³

Substitute:

V = 2/3 × 3.14 × 10³

Since:

10³ = 1,000

we get:

V ≈ 2,093.33 cm³

Therefore, the bowl can hold approximately 2,093.33 cubic centimeters of space, assuming it is a perfect hemisphere.

Hemisphere volume can be useful for estimating the capacity of bowls, domes, tanks, and other objects with a half-sphere shape.

Common Mistakes When Finding the Volume of a Hemisphere

1. Using the Sphere Formula

A common mistake is using:

V = 4/3πr³

for a hemisphere.

That formula gives the volume of a full sphere.

For a hemisphere, use:

V = 2/3πr³

A simple way to remember this is:

Hemisphere = half of a sphere

2. Forgetting to Convert Diameter to Radius

If a problem gives the diameter, do not substitute it directly for r.

For example, if:

d = 10 cm

then:

r = 5 cm

Using 10 instead of 5 in the formula would produce a much larger answer because the radius is cubed.

3. Forgetting to Cube the Radius

The formula is:

V = 2/3πr³

not:

V = 2/3πr

You need to calculate:

r × r × r

before completing the calculation.

4. Confusing Volume and Surface Area

Volume uses:

V = 2/3πr³

Surface area uses a different formula.

For example, the total surface area of a hemisphere is:

A = 3πr²

Do not use a surface-area formula when the question asks for volume.

5. Using the Wrong Units

Volume is measured in cubic units.

Correct:

452.16 cm³

Incorrect:

452.16 cm²

The exponent 3 is important because volume measures three-dimensional space.

FAQ

1. What is the formula for the volume of a hemisphere?

The formula is V = 2/3πr³, where r is the radius. It is exactly half of the volume of a sphere with the same radius.

2. How do you find the volume of a hemisphere?

Find the radius, cube it, multiply by π and 2, and divide by 3. In formula form: V = 2/3πr³.

3. How do you find the volume of a hemisphere when the diameter is given?

First divide the diameter by 2 to find the radius. Then substitute the radius into V = 2/3πr³. For example, a diameter of 12 cm gives a radius of 6 cm.

4. Is the volume of a hemisphere half the volume of a sphere?

Yes. A hemisphere is exactly half of a sphere when the sphere is divided through its center. Therefore, its volume is half of the sphere’s volume.

5. What is the difference between a sphere and a hemisphere?

A sphere is a complete three-dimensional round shape. A hemisphere is exactly half of a sphere, and has a curved surface plus a flat circular base. The volume of a hemisphere is half the volume of a sphere with the same radius.

Conclusion

The volume of a hemisphere measures the space inside half of a sphere. The key formula to remember is V = 2/3πr³. If the radius is given, substitute it directly into the formula. If the diameter is given, divide it by 2 first to find the radius. Because volume is three-dimensional, the final answer should always use cubic units.

The most important idea is that a hemisphere is exactly half of a sphere. Once you understand the sphere formula 4/3πr³, the hemisphere formula 2/3πr³ becomes much easier to remember and apply.

For students who want more guided practice with geometry, formulas, and problem-solving, WuKong Education provides structured online math class for kids, with practices for different grade levels.

James | WuKong Math Teacher

Graduated from Columbia University in the United States and has rich practical experience in mathematics competitions’ teaching, including Math Kangaroo, AMC 8… He teaches students the ways to flexible thinking and quick thinking in sloving math questions, and he is good at inspiring and guiding students to think about mathematical problems and find solutions.