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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 you’re still stuck.

A **cone **is a three dimensional object with a circular base that tapers to a point (a vertex) that is directly above the centre of the base.

This type of cone is sometimes referred to as a “right circular cone”.

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

The **perpendicular height** of a cone is h

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

The **volume** of a cone is how much space there is inside a cone.

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}**Step by step guide:** __Volume of a cone__ (coming soon)

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)}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\\\\ &= 129.6849… \end{aligned}

\begin{aligned}
\text{Area of circle }&=\pi r^2\\\\
&=\pi \times 4.3^2\\\\
&=58.0880…
\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\\\\ &=83.754…\\\\ &=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…\\\\ &=266.90…\\\\ &=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 \ cm^2

118 \ cm^2

116 \ cm^2

119 \ 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

- Volume of a cone
- Surface area of a cone
- Volume of a sphere
- Surface area of a sphere
- Volume of a pyramid
- 3D Pythagoras

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