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Here we will learn about spheres, including how to find the volume and surface area of a sphere.

There are also sphere* *worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

A **sphere shape** is three dimensional. Every point of its surface is equidistant from the center of the sphere. A sphere has no vertices.

A **hemisphere **is half a sphere.

The **volume** of a sphere is the amount of space there is inside a sphere.

The formula for the volume of a sphere is:

\text{Volume}=\frac{4}{3}\pi{r}^3E.g. Find the volume of the sphere

This sphere has a radius of 7 \; cm .

\begin{aligned} \text{Volume}&=\frac{4}{3} \pi r^3 \\\\ &= \frac{4}{3} \times \pi \times 7^3\\\\ &=\frac{1372}{3}\pi\\\\ &=1436.755... \\\\ &=1440 \ cm^3 \ \text{(3sf)} \end{aligned}**Step-by-step guide:** __Volume of a sphere__ (coming soon)

The **surface area** of a sphere is the area which covers the outer surface of a sphere.

The formula for the surface area of a sphere is:

\text{Surface Area}=4 \pi r^2E.g. Find the surface area of the sphere

This sphere has a radius of 7 \; cm .

\begin{aligned} \text{Surface area}&=4\pi r^2\\\\ &=4 \times \pi \times 7^2\\\\ &=196\pi\\\\ &=615.752...\\\\ &=616 \ cm^2 \ \text{(to 3 sf)} \end{aligned}**Step-by-step guide: **Surface area of a sphere

In order to calculate the volume of a sphere or the surface area of a sphere:

**Write down the formula.****Substitute the given values.****Work out the calculation****Write the final answer.**

Get your free sphere shape worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free sphere shape worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONCalculate the volume of a sphere of radius 12 \; cm. Give your answer to 3 significant figures.

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

2**Substitute the given values into the formula.**

3**Complete the calculation.**

4**Write the final answer, including the units.**

The radius is given in centimetres so the volume will be cubic centimetres.

The volume of the sphere is 7240 \; cm^3 (3 sf) .

Calculate the volume of a sphere with a diameter of 30 \; cm . Write your answer in terms of \pi .

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

\text{Volume}=\frac{4}{3} \pi r^3

**Substitute the given values into the formula.**

The diameter is twice the length of the radius. To calculate the radius, we need to first divide the diameter by 2:

30 \div 2=15 \; cm

The radius is 15 \; cm. Substituting this into the formula for the volume of a sphere, we have

V=\frac{4}{3} \times \pi \times 15^3

**Complete the calculation.**

V=4500\pi

**Write the final answer, including the units.**

The radius is given in centimetres so the volume will be cubic centimetres.

The volume of the sphere is 4500\pi \; cm^3 .

Calculate the volume of a hemisphere of radius 4 \; cm . Write your answer to 2 decimal places.

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

\text{Volume}=\frac{4}{3} \pi r^3

**Substitute the given values into the formula.**

V=\frac{4}{3} \times \pi \times 4^3

**Complete the calculation.**

V=268.0825...

But this is for a whole sphere. We need a hemisphere, so we need to divide by 2.

268.0825…\div 2=134.041...

** Write the final answer, including the units.**

The radius is given in centimetres so the volume will be cubic centimetres.

The volume of the hemisphere is 134.04 \; cm^3 \; (to \; 2 dp).

Calculate the surface area of a sphere with radius 7 \; cm . Write your answer to 1 decimal place.

**Write down the formula for the surface area of a sphere.**

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

**Substitute the given values into the formula.**

Here we know the radius of the sphere is 7 \; cm so we substitute r=7 into the formula for the surface area.

SA=4\times\pi\times{7}^{2}

**Complete the calculation.**

\begin{aligned} SA&=196\pi\\\\ &=615.7521601… \end{aligned}

**Write the final answer, including the units.**

The radius is measured in centimetres so the surface area is measured in square centimetres (cm^2) , with the solution rounded to 1 decimal place.

The surface area of the sphere is 615.8 \; cm^2 .

Find the surface area of a sphere with the diameter of 20 \; m. Leave your answer in terms of \pi .

**Write down the formula for the surface area of a sphere.**

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

**Substitute the given values into the formula.**

The diameter of the sphere is 20m and so we need to use this to calculate the radius and hence the surface area of the sphere.

20 \div 2=10

The radius of the sphere is 10m.

We now know the radius of the sphere is 10m so we substitute r=10 into the formula for the surface area.

SA=4\times\pi\times{10}^{2}

**Complete the calculation.**

SA=400\pi

**Write the final answer, including the units.**

The radius is measured in metres so the surface area is measured in square metres (m^2) , with the solution in terms of \pi .

The surface area of the sphere is 400\pi m^2 .

Find the total surface area of a hemisphere with a radius 4 \; cm. Write your answer to 3 significant figures.

**Write down the formula for the surface area of a sphere.**

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

For the surface area of a hemisphere we need half of the surface area of a sphere and add it on to the area of the flat circular base. The formula can be adapted.

\text{Total Surface Area}=\frac{1}{2}\times 4\pi r^2 + \pi r^2

**Substitute the given values into the formula.**

\text{Total SA}=2 \times \pi \times 4^2+\pi \times 4^2

**Complete the calculation.**

\begin{aligned} \text{Total SA}&=48\pi\\\\ &=150.796… \end{aligned}

**Write the final answer, including the units.**

The radius is measured in centimetres so the surface area is measured in square centimetres (cm^2) , with the solution rounded to 3 significant figures.

The surface area of the sphere is 151 \; cm^2 \; (to \; 3 sf) .

**Using the correct formula**

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

**Rounding**

It is important to not round the answer until the end of the calculation. This will mean your final answer is accurate.

**Make sure you have the correct units**

For area we use square units such as cm^2, m, km^2.

For volume we use cube units such as cm^3, m^3, km^3.

**Using the radius or the diameter**

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

1. Find the volume of a sphere with radius 4.3 \; cm. Write your answer to 2 decimal places.

333.04 \; cm^3

330 \; m^3

58.09 \; cm^3

232.35 \; cm^3

We are finding the volume of a sphere so we substitute the value of the radius r=4.3 into the formula V=\frac{4}{3}\pi{r}^{3}

\begin{aligned} \text{Volume}&=\frac{4}{3} \pi r^3 \\\\ &= \frac{4}{3} \times \pi \times 12^3\\\\ &=333.0381428… \\\\ &=333.04\text{cm}^{3} \ \text{(2dp)} \end{aligned}

2. Find the volume of a sphere with radius 18 \; cm. Leave your answer in terms of \pi .

432\pi \; cm^3

1944\pi \; m^3

7776\pi \; cm^3

5832\pi \; cm^3

We are finding the volume of a sphere so we substitute the value of the radius into the formula V=\frac{4}{3}\pi{r}^{3} . As the diameter is twice the length of the radius, r=36 \div 2=18 \;cm.

\begin{aligned} \text{Volume}&=\frac{4}{3} \pi r^3 \\\\ &= \frac{4}{3} \times \pi \times 18^3\\\\ &=7776\pi \\\\ &=7776\pi \; cm^3 \end{aligned}

3. Find the volume of a hemisphere with radius 6.2 \; cm. Write your answer to 1 decimal place.

362.3 \; cm^3

998.3 \; m^3

499.2 \; cm^3

120.8 \; cm^3

We are finding the volume of a hemisphere, so we need to calculate half of the volume of a sphere using the radius r=6.2 \; cm, and the formula: V=(\frac{4}{3}\pi{r}^3)\div{2}.

\begin{aligned} V&=(\frac{4}{3}\pi{r}^3)\div{2}\\\\ &=(\frac{4}{3}\times\pi\times{6.2}^3)\div{2}\\\\ &=499.152996…\\\\ &=499.2\text{ cm}^3 \end{aligned}

4. Find the surface area of a sphere with radius 4.6 \; cm. Give your answer to 2 decimal places.

270\text{ cm}^2

407.72\text{ cm}^2

66.48\text{ cm}^2

265.90\text{ cm}^2

We are finding the surface area of a sphere, so we substitute the value of the radius r=4.6 into the formula SA=4\pi r^2.

\begin{aligned} \text{Surface area}&=4\pi r^2\\\\ &=4 \times \pi \times 4.6^2\\\\ &=265.9044022…\\\\ &=265.90 \text{ cm}^2 \ \text{(2dp)} \end{aligned}

5. Find the surface area of a sphere with a diameter of 16 \; mm. Leave your answer in terms of \pi.

1024\pi\text{ mm}^2

64\pi\text{ mm}^2

256\pi \text{ mm}^2

2144\pi\text{ mm}^2

We are finding the surface area of a sphere, so we need to calculate the value of the radius using the diameter, then substitute the value of the radius into the formula for the surface area.

r=16 \div 2=8 \; mm

The radius r=8 \; mm.

\begin{aligned} \text{Surface area}&=4\pi r^2\\\\ &=4 \times \pi \times 8^2\\\\ &=256\pi \ \text{mm}^2 \end{aligned}

6. Find the surface of a hemisphere with radius 7.8 \; cm. Give your answer to 1 decimal place.

573.4 \text{ cm}^2

764.5 \text{ cm}^2

191.1 \text{ cm}^2

993.9 \text{ cm}^2

We need to find the curved surface area by finding half of the surface area of a sphere. Then we add on the area of the circular top.

\begin{aligned} \text{Total surface area}&=3\pi r^2\\\\ &=3 \times \pi \times{7.8}^2\\\\ &=573.4034911…\\\\ &=573.4 \text{ cm}^2 \ \text{(1dp)} \end{aligned}

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

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

1. Here is a sphere with a radius 8.4 \; cm.

Calculate the volume of the sphere.

Write your answer to 1 decimal place.

**(2 marks)**

Show answer

V=\frac{4}{3} \times \pi \times 8.4^3

**(1)**

2482.71271…=2482.7 \; cm^3

**(1)**

2. A tennis ball has a radius of 3.4 \; cm. Each tennis ball is covered in felt. Calculate the exact area of the felt required to cover the ball to the nearest square millimetre.

**(4 marks)**

Show answer

3.4 \; cm=34 \; mm

**(1)**

SA=4\times \pi \times{34}^2

**(1)**

14526.72443…

**(1)**

14527 \; mm^2

**(1)**

3. A container is a hemisphere with a diameter of 70 \; cm.

Sand fills the container at a rate of 5000 \; cm^3 per minute.

How long does it take to fill the container?

Give your answer to the nearest minute.

**(3 marks)**

Show answer

(\frac{4}{3} \times \pi \times 35^3) \div{2} = 89797.19002…

**(1)**

89797.19002… \div{5000}=17.959438…

**(1)**

17.959438… = 18 minutes

**(1)**

4. A sphere has a surface area of 1000 \; m^2.

Calculate its volume.

Give your answer to 3 significant figures.

**(3 marks)**

Show answer

r=\sqrt{\frac{1000}{4\pi}}=8.920620581…

**(1)**

\frac{4}{3}\times \pi \times8.920620581^3=2973.540194…

**(1)**

2973.540194…=2970 \; m^3 \; (3sf)

**(1)**

You have now learned how to:

- Find the volume of a sphere
- Find the surface area of a sphere
- How to solve problems involving spheres

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