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Multiplying fractions Area of a circle Rounding Sphere shape Volume of a sphere Surface area of a sphereHere you will learn about the hemisphere shape, including what a hemisphere is and how to calculate the volume of a hemisphere and the surface area of a hemisphere.

Students will first learn about the hemisphere shape as part of geometry in elementary school, but they will learn calculations involving the hemisphere shape in high school.

A **hemisphere shape** is a three-dimensional shape that is half of a sphere.

To visualize this, let’s look at a sphere. A sphere is a 3D shape where every point of its surface is equidistant (the same distance) from the center of the sphere. It has a radius r.

A hemisphere is half of a sphere. The prefix “hemi” means half.

You may have come across the word hemisphere in the real life context of planet Earth. You can split the world into 2 equal halves – the northern hemisphere and the southern hemisphere.

Use this quiz to check your grade 1, 5 and 6 students’ understanding of 3D shape. 10+ questions with answers covering a range of 1st, 5th and 6th grade 3D shape topics to identify areas of strength and support!

DOWNLOAD FREEUse this quiz to check your grade 1, 5 and 6 students’ understanding of 3D shape. 10+ questions with answers covering a range of 1st, 5th and 6th grade 3D shape topics to identify areas of strength and support!

DOWNLOAD FREEA hemisphere is a geometric shape which has a curved surface area. The base of a hemisphere is a flat face, which is a circle.

A hemisphere has no vertices and one edge. It is not made up of polygons, so it is not a polyhedron.

Faces: 2

Edges: 1

Vertices: 0

The **volume of a hemisphere** is based on the volume of a sphere.

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

\text{Volume of a sphere}=\cfrac{4}{3} \pi r^3

To find the volume of a hemisphere you halve the volume of a sphere.

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

\text{Volume of a hemisphere}=\cfrac{4}{3} \pi r^3\div 2

or

\text{Volume of a hemisphere}=\cfrac{2}{3} \pi r^{3}

**See also:** Volume of a hemisphere

The **surface area of a hemisphere** is based on the surface area of a sphere.

To calculate the surface area of a sphere with radius r, you can use the formula

\text{Surface area of a sphere}=4\pi{r}^{2}

So to find the curved surface area of the hemisphere, you need to halve the surface area of the sphere.

4\pi{r}^{2}\div 2=2\pi{r}^{2}

However this would only give the curved surface area. If you want the total surface area, you need to add the area of the base of the hemisphere.

The area of the base, which is a circle, is given by \pi r^{2}.

Here is the **surface area of a hemisphere formula**, for a hemisphere with radius r

\text{Total surface area of a hemisphere}=2\pi{r}^{2}+\pi{r}^{2}=3\pi{r}^{2}.

**See also:** Surface area of a hemisphere

How does this relate to high school math?

**High School – Geometry – Geometric Measurement & Dimension (HS.G.GMD.A.2)**Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

In order to find the volume or surface area of a hemisphere:

**Write down the formula for the sphere.****Adapt the formula for the question.****Substitute in the value.****Write the final answer.**

The radius of a hemisphere is 12.4\mathrm{~cm}. Find the volume of the hemisphere to the nearest whole number.

**Write down the formula for the sphere.**

You need to consider the formula for the volume of a sphere.

V=\cfrac{4}{3} \pi r^32**Adapt the formula for the question.**

The formula needs to be halved for the volume of a hemisphere.

V=\cfrac{4}{3} \pi r^3 \div 23**Substitute in the value.**

Then you substitute in the value 12.4 in place of r.

V=\cfrac{4}{3} \pi (12.4)^3 \div 24**Write the final answer.**

The answer is 3993.223…

This rounds to give 3993 \mathrm{~cm}^3 (to the nearest whole number) as the volume of the hemisphere.

Find the volume of a hemisphere with radius 15\mathrm{~cm}. Leave your answer in terms of \pi.

**Write down the formula for the sphere.**

You need to consider the formula for the volume of a sphere.

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

**Adapt the formula for the question.**

The formula needs to be halved for the volume of a hemisphere.

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

**Substitute in the value.**

Then you substitute in the value 15 in place of r.

V=\cfrac{4}{3} \pi (15)^3 \div 2

**Write the final answer.**

The volume of the hemisphere is 2250\pi \mathrm{~cm}^{3}.

Find the curved surface area of a hemisphere with diameter 4.5\mathrm{~m}. Give your answer to the nearest tenth.

**Write down the formula for the sphere.**

You need to consider the formula for the surface area of a sphere.

\text{Surface Area}=4\pi{r}^{2}

**Adapt the formula for the question.**

The formula needs to be halved for the curved surface area of a hemisphere.

\text{Curved Surface Area}=4\pi{r}^{2}\div 2=2\pi{r}^{2}

**Substitute in the value.**

You have been given a diameter 4.5. You need to divide the diameter by 2 to get the radius. So you substitute in the value 2.25 in place of r.

\text{Curved Surface Area}=2\pi{(2.25)}^{2}

**Write the final answer.**

The answer is 31.808…

This rounds to give 31.8 \mathrm{~m}^2 (to the nearest tenth) as the curved surface area of the hemisphere.

Find the total surface area of a hemisphere with radius 7.6\mathrm{~cm}. Give your answer to the nearest whole number.

**Write down the formula for the sphere.**

You need to consider the formula for the surface area of a sphere.

\text{Surface Area}=4\pi{r}^{2}

**Adapt the formula for the question.**

The formula needs to be halved for the curved surface area of a hemisphere.

\text{Curved Surface Area}=4\pi{r}^{2}\div 2=2\pi{r}^{2}

But, you need to also add the area of the circular base to get the total surface area of the hemisphere.

\text{Total Surface Area}=2\pi{r}^{2}+\pi{r}^{2}=3\pi{r}^{2}

**Substitute in the value.**

Then you substitute in the value 7.6 in place of r.

\text{Total Surface Area}=3\pi{(7.6)}^{2}

**Write the final answer.**

The answer is 544.375…

This rounds to give 544 \mathrm{~cm}^2 (to the nearest whole number) as the total surface area of the hemisphere.

Find the curved surface area of a hemisphere with radius 8\mathrm{~cm}. Leave your answer in terms of \pi.

**Write down the formula for the sphere.**

You need to consider the formula for the surface area of a sphere.

\text{Surface Area}=4\pi{r}^{2}

**Adapt the formula for the question.**

The formula needs to be halved for the curved surface area of a hemisphere.

\text{Curved Surface Area}=4\pi{r}^{2}\div 2=2\pi{r}^{2}

**Substitute in the value.**

Then you substitute in the value 8 in place of r.

\text{Curved Surface Area}=2\pi{(8)}^{2}

**Write the final answer.**

The curved surface area of the hemisphere is 128\pi \mathrm{~cm}^{2}.

Find the total surface area of a hemisphere with diameter 10\mathrm{~cm}. Leave your answer in terms of \pi.

**Write down the formula for the sphere.**

You need to consider the formula for the surface area of a sphere.

\text{Surface Area}=4\pi{r}^{2}

**Adapt the formula for the question.**

The formula needs to be halved for the curved surface area of a hemisphere.

\text{Curved Surface Area}=4\pi{r}^{2}\div 2=2\pi{r}^{2}

But, you need to also add the area of the circular base to get the total surface area of the hemisphere.

\text{Total Surface Area}=2\pi{r}^{2}+\pi{r}^{2}=3\pi{r}^{2}

**Substitute in the value.**

You have been given a diameter 10. You need to divide the diameter by 2 to get the radius. So you substitute in the value 5 in place of r.

\text{Total Surface Area}=3\pi{(5)}^{2}

**Write the final answer.**

The total surface area of the hemisphere is 75 \pi \mathrm{~cm}^2.

- Use physical models of spheres and hemispheres. Show everyday items like hemispherical bowls, domes, and igloos.

- Incorporate problem-solving exercises and worksheets that require students to apply the formulas for volume of a hemisphere and surface area of a hemisphere in different contexts.

- Hands-on activity: Have students create both solid and hollow hemisphere models using materials like clay for solid hemispheres and papier-mâché or plastic for hollow hemispheres. Let students weigh the solid and hollow models to understand the difference in mass. Have them calculate the volume and surface area for each type, reinforcing the mathematical concepts.

**Using the incorrect formula**

There are several formulas that can be used in geometry, so you need to match the correct formula to the correct context.

**Mixing up total surface area and curved surface area**

Double check whether the question requires you to find the curved surface area of a hemisphere or the total surface area of a hemisphere.

**Rounding too soon**

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 incorrect units**

For area you use square units such as \mathrm{cm}^2.

For volume you use cube units such as \mathrm{cm}^3.

**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.

- 3D shapes
- 3D shape names
- Pyramid shape
- Square pyramid
- Triangular pyramid
- Cylinder
- Cone
- Angles of elevation and depression

1. Find the volume of a hemisphere with radius 11.3\mathrm{~cm}. Give your answer to the nearest whole number.

3020 \mathrm{~cm}^3

3022 \mathrm{~cm}^3

6044 \mathrm{~cm}^3

6040 \mathrm{~cm}^3

You need to halve the formula for the volume of a sphere, and then substitute the value of 11.3 in place of r.

\begin{aligned}V&=\cfrac{4}{3} \pi {r}^{3}\div 2 \\\\ V&=\cfrac{4}{3} \times \pi \times {11.3}^{3}\div 2 \\\\ V&=3021.996… \\\\\ V&=3022 \mathrm{~cm}^3 \\ & \text{(to the nearest whole number)} \end{aligned}

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

\cfrac{16}{3}\pi \mathrm{~cm}^3

\cfrac{32}{3}\pi \mathrm{~cm}^3

\cfrac{256}{3}\pi \mathrm{~cm}^3

\cfrac{128}{3}\pi \mathrm{~cm}^3

You have been given the diameter 4, which you need to divide by 2 to get the radius. You need to halve the formula for the volume of a sphere, and then substitute the value of 2 in place of r.

\begin{aligned}V&=\cfrac{4}{3} \pi {r}^{3}\div 2 \\\\ V&=\cfrac{4}{3} \times \pi \times {2}^{3}\div 2 \\\\ V&=\cfrac{16}{3}\pi \\\\ V&=\cfrac{16}{3}\pi \mathrm{~cm}^3 \end{aligned}

3. Find the curved surface area of a hemisphere with radius 12.8\mathrm{~cm}. Give your answer to the nearest whole number.

515 \mathrm{~cm}^2

1029 \mathrm{~cm}^2

1540 \mathrm{~cm}^2

2060 \mathrm{~cm}^2

You need to halve the formula for the surface area of a sphere to find the curved surface area of a hemisphere (CSA). Then you substitute the value of 12.8 in place of r.

\begin{aligned}CSA&=4 \pi {r}^{2}\div 2 \\\\ CSA&=2\pi {r}^{2} \\\\ CSA&=2\times \pi \times {12.8}^{2} \\\\ CSA&=1029.437… \\\\ CSA&=1029 \mathrm{~cm}^2 \\ & \text{(to the nearest whole number)} \end{aligned}

4. Find the curved surface area of a hemisphere with radius 11\mathrm{~cm}. Leave your answer in terms of \pi.

242\pi \mathrm{~cm}^2

121\pi \mathrm{~cm}^2

968\pi \mathrm{~cm}^2

44\pi \mathrm{~cm}^2

You need to halve the formula for the surface area of a sphere to find the curved surface area of a hemisphere (CSA). Then you substitute the value of 11 in place of r.

\begin{aligned}CSA&=4 \pi {r}^{2}\div 2 \\\\ CSA&=2\pi {r}^{2} \\\\ CSA&=2\times \pi \times {11}^{2} \\\\ CSA&=242\pi \\\\ CSA&=242\pi \mathrm{~cm}^2 \end{aligned}

5. Find the total surface area of a hemisphere with radius 62\mathrm{~cm}. Give your answer to the nearest hundred.

48400 \mathrm{~cm}^2

48300 \mathrm{~cm}^2

36300 \mathrm{~cm}^2

36200 \mathrm{~cm}^2

You need to halve the formula for the surface area of a sphere and add on the circular base to find the total surface area of a hemisphere (TSA). Then you substitute the value of 62 in place of r.

\begin{aligned}TSA&=4 \pi {r}^{2}\div 2 + 2\pi {r}^{2} \\\\ TSA&=3\pi {r}^{2} \\\\ TSA&=3\times \pi \times {62}^{2} \\\\ TSA&=36228.8… \\\\ TSA&=36300 \mathrm{~cm}^2 \\ & \text{(to the nearest hundred)} \end{aligned}

6. Find the total surface area of a hemisphere with radius 15\mathrm{~cm}. Leave your answer in terms of \pi.

225\pi \mathrm{~cm}^2

450\pi \mathrm{~cm}^2

675\pi \mathrm{~cm}^2

900\pi \mathrm{~cm}^2

You need to halve the formula for the surface area of a sphere and add on the circular base to find the total surface area of a hemisphere (TSA). Then you substitute the value of 15 in place of r.

\begin{aligned}TSA&=4 \pi {r}^{2}\div 2 + 2\pi {r}^{2} \\\\ TSA&=3\pi {r}^{2} \\\\ TSA&=3\times \pi \times {15}^{2} \\\\ TSA&=675\pi \\\\ TSA&=675\pi \mathrm{~cm}^2 \, \text{(to 3 sf)} \end{aligned}

A hemisphere is a three-dimensional shape that represents half of a sphere. It is formed by cutting a sphere into two equal halves along a plane passing through its center.

The total surface area of a hemisphere includes both the curved surface area and the area of the flat circular base, while the curved surface area (also called lateral surface area) includes only the curved part of the hemisphere.

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