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

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

A **cone **is a three dimensional object with a circular base that tapers to a point that is directly above the centre of the base. This point is called the vertex or apex of the cone.

Cone shapes that you are used to in real-life would be ice cream cones or traffic cones. This type of cone is sometimes referred to as a βright circular coneβ or “right cone”. There are also oblique cones where the apex is not directly above the centre of the base, and also cones that have an ellipse as a base rather than a circle.

Here are the important properties of a cone that you will need for GCSE Mathematics.

The **radius **of the base of the cone is r

The **perpendicular height** of the cone is h , this is the height at a right angle to the base.

The **slant height** of the cone is l

Cones feature in A level Further Maths in a topic called Conic Sections. This topic explores the links between cones and other shapes depending on the angle that a cone is sliced.

The **volume** of a cone is how much space there is inside a cone. The volume of a cone is one third of the volume of a cylinder with the same height and radius.

The formula for the volume of a cone is:

\text{Volume}=\frac{1}{3} \pi r^2 hE.g. Find the volume of the cone

\begin{aligned} \text{Volume of a cone}&=\frac{1}{3}\pi r^2 h\\\\ &=\frac{1}{3} \times \pi \times 3^2 \times 4\\\\ &=12\pi \\\\ &= 37.7 \ cm^3 \ \text{(to 1 dp)} \end{aligned}You also need to find the volume of a frustum. A frustum is the part of a cone left when the top part has been cut off.

**Step-by-step guide:** Volume of a cone

In order to calculate the volume of a cone:

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

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

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

COMING SOONFind the volume of the cone with radius 5.3 \ cm and perpendicular height 7.8 \ cm .

Give your answer to 3 significant figures.

**Write down the formula.**

2**Substitute the given values.**

3**Work out the calculation.**

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

Find the volume of the cone with radius 9 \ cm and perpendicular height 11 \ cm .

Leave your answer in terms of \pi .

**Write down the formula.**

V \text { olume } =\frac{1}{3} \pi r^{2} h

**Substitute the given values.**

\begin{aligned}
V \text { olume } &=\frac{1}{3} \pi r^{2} h \\\\
&=\frac{1}{3} \times \pi \times 9^{2} \times 11
\end{aligned}

**Work out the calculation.**

\begin{aligned}
&=\frac{1}{3} \times \pi \times 9^{2} \times 11 \\\\
&=297 \pi
\end{aligned}

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

The question asks for the answer in terms of \pi so the final answer is 297\pi \ cm^3

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

The surface area is made up of two parts, a curved surface area and a circular base.

The formula for calculating the **curved surface area of a cone** is:

The formula for calculating the **area of a circle**:

For the **total surface area**, we can add the two parts together:

E.g.

\text{Curved surface area}=\pi rl=\pi \times 3\times 5=15\pi \text{Area of circle}=\pi r^2=\pi \times 3^2=9\pi \text{total surface area}=15\pi +9\pi =24\pi=75.4 cm^2 \ \text{(to 1 dp)}**Step-by-step guide:** Surface area of a cone

In order to calculate the surface area of a cone:

**Work out the area of each face.****Add the areas together.****Include units.**

Find the curved surface area of the cone with radius 4.3 \ cm and slant height 9.6 \ cm.

Give your answer to 3 significant figures.

**Work out the area of each face**

\begin{aligned} \text{Curved surface area}&=\pi rl\\\\ &=\pi \times 4.3 \times 9.6\\\\ &= 129.6849β¦ \end{aligned}

\begin{aligned}
\text{Area of circle }&=\pi r^2\\\\
&=\pi \times 4.3^2\\\\
&=58.0880β¦
\end{aligned}

**Add the areas together.**

Total surface area: 129.6849+58.0880=187.7729...

**Include units.**

Surface area =188cm^2 \ (3sf)

Find the curved surface area of the cone with radius 8 \ cm and slant height 13 \ cm.

Leave your answer in terms of \pi .

**Work out the area of each face**

\begin{aligned}
\text{Curved surface area}&=\pi rl\\\\
&=\pi \times 8 \times 13\\\\
&= 104 \pi
\end{aligned}

\begin{aligned}
\text{Area of circle }&=\pi r^2\\\\
&=\pi \times 8^2\\\\
&=64\pi
\end{aligned}

**Add the areas together.**

Total surface area: 104\pi +64\pi = 168\pi

**Include units.**

=168 \pi \mathrm{cm}^{2}

**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.

**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 and the diameter is double the radius.

**Make sure you have the correct units**

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

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

1. Find the volume of a cone of radius 9.4 \ cm and perpendicular height 8.7 \ cm

Give your answer to 3 significant figures.

805 \ cm^3

806 \ cm^3

745 \ cm^3

746 \ cm^3

We are finding the volume of a cone so we substitute the value of r and h into the formula.

\begin{aligned} V&=\frac{1}{3} \pi r^2 h\\\\ V&=\frac{1}{3}\times \pi \times 9.4^2 \times 8.7\\\\ V&=805.014…\\\\ V&=805 \ cm^3 \ \text{(to 3 sf)} \end{aligned}

2. Find the volume of a cone of radius 8 \ cm and perpendicular height 6 \ cm

Leave your answer in terms of \pi .

127\pi \ cm^3

126\pi \ cm^3

128\pi \ cm^3

125\pi \ cm^3

We are finding the volume of a cone so we substitute the value of r and h into the formula.

\begin{aligned} V&=\frac{1}{3} \pi r^2 h\\\\ V&=\frac{1}{3}\times \pi \times 8^2 \times 6\\\\ V&=128\pi\\\\ V&=128\pi \ cm^3 \end{aligned}

3. Find the curved surface area of a cone of radius 4.3 \ cm and slant height 6.2 \ cm.

Give your answer to 1 decimal place.

83.7 \ cm^2

360.1 \ cm^2

360.2 \ cm^2

83.8 \ cm^2

We are finding the curved surface area of a cone so we substitute the value of r and l into the formula.

\begin{aligned} \text{Curved surface area}&=\pi rl\\\\ &=\pi \times 4.3\times 6.2\\\\ &=83.754β¦\\\\ &=83.8 \ cm^2 \ \text{(to 1 dp)} \end{aligned}

4. Find the curved surface area of a cone of radius 7 \ cm and slant height 9 \ cm.

Leave your answer in terms of \pi .

61\pi \ cm^2

63\pi \ cm^2

62\pi \ cm^2

64\pi \ cm^2

We are finding the curved surface area of a cone so we substitute the value of r and l into the formula.

\begin{aligned} \text{Curved surface area}&=\pi rl\\\\ &=\pi \times 7\times 9\\\\ &=63\pi\\\\ &=63\pi \ cm^2 \end{aligned}

5. Find the** total surface area** of a cone of radius 5.9 \ cm and slant height 8.5 \ cm.

Give your answer to 3 significant figures.

266 \ cm^2

267 \ cm^2

384 \ cm^2

385 \ cm^2

We are finding the **total surface area** of a cone so we find the curved surface area and add on the area of the circular base.

\begin{aligned} \text{TOTAL surface area}&=\pi rl+\pi r^2\\\\ &=\pi \times 5.9\times 8.5 + \pi \times 5.9^2\\\\ &=157.5508… + 109.3588…\\\\ &=266.90β¦\\\\ &=267 \ cm^2 \ \text{(to 3 sf)} \end{aligned}

6. Find the** total surface area** of a cone of radius 7 \ cm and slant height 10 \ cm.

Leave your answer in terms of \pi.

117\pi \ cm^2

118\pi \ cm^2

116\pi \ cm^2

119\pi \ cm^2

Make sure you find the curved surface area and the area of the circular base

\begin{aligned} \text{total surface area}&=\pi rl+\pi r^2\\\\ &=\pi \times 7\times 10 + \pi \times 7^2\\\\ &=70\pi + 49\pi\\\\ &=119\pi\\\\ &=119\pi \ cm^2 \end{aligned}

\text{Volume of a cone}=\frac{1}{3} \pi r^2 h
\text{Curved surface area of a cone}=\pi rl

1. Here is a cone with radius 7.3 \ cm and perpendicular height 9.5 \ cm.

Find the volume of the cone.

Give your answer to 3 significant figures.

**(2 marks)**

Show answer

V=\frac{1}{3} \times \pi \times 7.3^2 \times 9.5

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2. Here is a cone.

Find the** total surface area** of the cone.

Give your answer to 3 significant figures.

**(3 marks)**

Show answer

= \pi \times 14 \times 19=835.663…

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3. A container is a cone of radius 60 \ cm and perpendicular height 80 \ cm.

Water fills the container at a rate of 9000 \ 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{1}{3} \times \pi \times 60^2 \times 80= 301 592.89…

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4. Here is a cone.

Find the volume of the cone..

Give your answer to 3 significant figures.

**(3 marks)**

Show answer

h=\sqrt{10^2 – 4^2}=9.1651…

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5. A cone has a radius of 11 \ cm.

It has a volume of 2000 \ cm^3.

Find the **total surface area** of the cone.

Give your answer to 3 significant figures.

**(5 marks)**

Show answer

h=\frac{3\times 2000}{11^2 \times \pi}=15.783…

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You have now learned how to:

- Calculate the volume of a cone
- Calculate the curved surface area of a cone
- Calculate the total surface area of a cone

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