# 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

### What is a cone? ## 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.

### How to calculate the volume of a cone ## 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 .

Give your answer to 3 significant figures.

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 .

Leave your answer in terms of \pi .

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.
2. Add the areas together.
3. Include units.

### How to calculate the surface area of a cone ## 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.

Give your answer to 3 significant figures.

\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.

Leave your answer in terms of \pi .

\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

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\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

### 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.

Give your answer to 3 significant figures.

(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.

Give your answer to 3 significant figures.

(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?

Give your answer to the nearest minute.

(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..

Give your answer to 3 significant figures.

(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.

Give your answer to 3 significant figures.

(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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#### GCSE Maths Papers - November 2022 Topics

Practice paper packs based on the November advanced information for Edexcel 2022 Foundation and Higher exams.

Designed to help your GCSE students revise some of the topics that are likely to come up in November exams.