# Cone

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.

## What is a cone?

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

## Volume of a cone

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 h

E.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)

## How to calculate the volume of a cone

In order to calculate the volume of a cone:

1. Write down the formula.
2. Substitute the given values.
3. Work out the calculation.
4. Write the final answer, including the units.

## Volume of a cone examples

### Example 1: volume of a cone

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

1. Write down the formula.

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

2Substitute the given values.

\begin{aligned} \text { Volume } &=\frac{1}{3} \pi r^{2} h \\\\ &=\frac{1}{3} \times \pi \times 5.3^{2} \times 7.8 \end{aligned}

3Work out the calculation.

\begin{aligned} &=\frac{1}{3} \times \pi \times 5.3^{2} \times 7.8 \\\\ &=229.443 \ldots \end{aligned}

4Write the final answer, including the units.

=229 \ cm^3 \ (to \ 3.s.f)

### Example 2: volume of a cone

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

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

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

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

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

## Surface area of a cone

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:

\text{Curved surface area}=\pi rl

The formula for calculating the area of a circle:

\text{Area of circle}=\pi r^2

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

\text{total surface area}=\pi rl+\pi r^2

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)}

## How to calculate the surface area of a cone

In order to calculate the surface area of a cone:

1. Work out the area of each face.
3. Include units.

## Surface area of a cone examples

### Example 3: total surface area of a cone

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

\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}

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

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

### Example 4: total surface area of a cone

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

\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}

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

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

### Common misconceptions

• 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.

### Practice cones questions

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

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

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.

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.

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.

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.

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

### Cone GCSE questions

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

Find the volume of the cone.

(2 marks)

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

(1)

530.148…=530

(1)

2. Here is a cone.

Find the total surface area of the cone.

(3 marks)

= \pi \times 14 \times 19=835.663…

(1)

835.663…+\pi \times 14^2

(1)

1451.4…=1450

(1)

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?

(3 marks)

\frac{1}{3} \times \pi \times 60^2 \times 80= 301 592.89…

(1)

301 592.89… \div 3000=33.51…

(1)

33.51…= 34 \ \text{minutes}

(1)

4. Here is a cone.

Find the volume of the cone..

(3 marks)

h=\sqrt{10^2 – 4^2}=9.1651…

(1)

V=\frac{1}{3} \times \pi \times 4^2 \times 9.1651…

(1)

153.563…=154

(1)

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.

(5 marks)

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

(1)

l=\sqrt{11^2 + 15.783…^2}=19.2388…

(1)

CSA= \pi \times 11\times 19.2388…=664.847…

(1)

664.847…+\pi \times 11^2

(1)

1044.97…=1040

(1)

## Learning checklist

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

## The next lessons are

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

## Still stuck?

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