Volume of a cone

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GCSE Maths Geometry and Measure 3D Shapes Cone

Volume Of A Cone

Here we will learn about the volume of a cone, including how to calculate the volume of a cone given its radius and perpendicular height and how to calculate a missing length within a cone, given its volume.

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 the volume of a cone?

The volume of a cone is the amount of space inside a cone.

The volume of a cone is one third of the volume of a cylinder.

The volume of a cone can be calculated using a formula.

To do this, we substitute two of the dimensions of the cone into the volume formula and evaluate the result.

The volume formula for any cone is

$katex is not defined$

where $katex is not defined$ is the radius of the base of the cone and $katex is not defined$ is the perpendicular height of the cone.

What is the volume of a cone?

Example

Find the volume of this cone where the radius of the base of the cone is $katex is not defined$ and the perpendicular height of the cone is $katex is not defined$ .

We need to find the area of the base of a cone using $katex is not defined$ (the area of a circle).

We then multiply it by the vertical height $katex is not defined$ , and find one third of the answer.

\[\text{Volume}=\frac{1}{3} \pi r^2 h\\ = \frac{1}{3} \times \pi \times 5^2 \times 8\\ =\frac{200}{3}\pi\\ =209.4 \ cm^3 \ \text{(1 dp)}\]

Types of cones

A cone is a three dimensional object with a circular base that tapers to a point (a vertex). This vertex is also known as the apex.

If the apex is directly above the centre of the base, the cone is referred to as a right circular cone. If the apex is not above the centre of the base, this is referred to as an oblique cone.

Note: The slant height of the cone l is not needed to calculate the volume of a cone, but it is used to calculate the surface area of a cone.

Step-by-step guide: Surface area of a cone (coming soon)

The volume of a cone stays the same if the apex moves parallel to the base but the surface area of the cone changes. This is because the slant heights are not equal for an oblique cone.

How to calculate the volume of a cone

In order to calculate the volume of a cone:

Write down the formula, $katex is not defined$ .
Substitute the given values.
Work out the calculation.
Write the final answer, including the units.

How to calculate the volume of a cone

Volume of a cone worksheet

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

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Volume of a cone worksheet

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

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Related lessons on cones

Volume of a cone is part of our series of lessons to support revision on cone. You may find it helpful to start with the main cone lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Volume of a cone examples

Example 1: integer dimensions

Find the volume of the cone with radius $katex is not defined$ and perpendicular height $katex is not defined$ .

Give your answer to $katex is not defined$ significant figures.

Write down the formula, $katex is not defined$ .

The formula for the volume of a cone is $katex is not defined$ .

2Substitute the given values.

We need to substitute the value of the radius $katex is not defined$ and the perpendicular height $katex is not defined$ into the formula.

katex is not defined

3Work out the calculation.

Use a calculator to work out the volume.

katex is not defined

4Write the final answer, including the units.

Check what form the final answer needs to be. Here we are asked to give the answer to $katex is not defined$ significant figures.

katex is not defined

The volume of the cone is $katex is not defined$ .

Example 2: decimal dimensions

Find the volume of the cone with radius $katex is not defined$ and perpendicular height $katex is not defined$ .

Give your answer to $katex is not defined$ significant figures.

Write down the formula, $katex is not defined$ .

The volume formula of a cone is $katex is not defined$ .

Substitute the given values.

We need to substitute the value of the radius $katex is not defined$ and the perpendicular height $katex is not defined$ into the formula.

katex is not defined

Work out the calculation.

Use a calculator to work out the volume.

katex is not defined

Write the final answer, including the units.

Check what form the final answer needs to be. Here we are asked to give the answer to $katex is not defined$ significant figures.

katex is not defined

The volume of the cone is $katex is not defined$ .

Example 3: calculate the volume in terms of pi

Calculate the volume of a cone with radius $katex is not defined$ and perpendicular height $katex is not defined$ .

Write your answer in terms of $katex is not defined$ .

Write down the formula, $katex is not defined$ .

The formula for the volume of a cone is $katex is not defined$ .

Substitute the given values.

We need to substitute the value of the radius $katex is not defined$ and the perpendicular height $katex is not defined$ into the formula.

katex is not defined

Work out the calculation.

As we can change the order of a multiplication and the solution stays the same, we have $katex is not defined$ .

Simplifying this, we get $katex is not defined$ .

Write the final answer, including the units.

Check what form the final answer needs to be. Here we need to leave the answer in terms of $katex is not defined$ .

katex is not defined

The volume of the cone is $katex is not defined$ .

Example 4: calculate the volume given the diameter

Find the volume of the cone with the diameter of $katex is not defined$ and perpendicular height $katex is not defined$ .

Write your answer in terms of $katex is not defined$ .

Write down the formula, $katex is not defined$ .

The formula for the volume of a cone is $katex is not defined$ .

Substitute the given values.

As the diameter is twice the length of the radius, the radius is equal to $katex is not defined$ .

Now we can substitute the values of $katex is not defined$ and $katex is not defined$ into the formula to get $katex is not defined$ .

Work out the calculation.

As we can change the order of a multiplication and the solution stays the same, we have $katex is not defined$ .

Simplifying this, we get $katex is not defined$ .

Write the final answer, including the units.

Check what form the final answer needs to be. Here we are asked to leave the answer in terms of $katex is not defined$ .

katex is not defined

The volume of the cone is $katex is not defined$ .

Example 5: calculate a length given the volume

A cone has a volume of $katex is not defined$ and a radius of $katex is not defined$ . Calculate the perpendicular height of the cone, correct to $katex is not defined$ decimal place.

Write down the formula, $katex is not defined$ .

The volume formula of a cone is $katex is not defined$ .

Substitute the given values.

We are given the volume and the radius, so we can substitute these into the formula.

katex is not defined

As $katex is not defined$ , we have $katex is not defined$ .

Work out the calculation.

We need to solve the equation for $katex is not defined$ ,

Write the final answer, including the units.

Check what form the final answer needs to be. Here we are asked to give the answer to $katex is not defined$ decimal place.

katex is not defined

The perpendicular height of the cone is $katex is not defined$ .

Example 6: calculate the radius given the volume

A cone has a volume of $katex is not defined$ and a perpendicular height of $katex is not defined$ . Calculate the radius of the cone. Give your answer to $katex is not defined$ decimal places.

Write down the formula, $katex is not defined$ .

The question involves the volume of a cone, so we need the formula for the volume of a cone.

katex is not defined

Substitute the given values.

We are given the volume and the perpendicular height, so we can substitute these into the formula.

katex is not defined

Work out the calculation.

As $katex is not defined$ , we can simplify the right hand side of the equation, before we find the value of $katex is not defined$ .

We now have $katex is not defined$ . To find the value of $katex is not defined$ , we must square root both sides of the equation. By doing this we get $katex is not defined$

Write the final answer, including the units.

Check what form the final answer needs to be. Here we are asked to give the answer to $katex is not defined$ decimal places.

katex is not defined

The radius of the cone is $katex is not defined$ .

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.

Make sure you have the correct units
- For area we use square units such as $katex is not defined$ .
- For volume we use cube units such as $katex is not defined$ .

Practice volume of a cone questions

katex is not defined

katex is not defined

katex is not defined

katex is not defined

Substitute the values of $katex is not defined$ and $katex is not defined$ into the formula.

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

Substitute the values of $katex is not defined$ and $katex is not defined$ into the formula.

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

We are finding the volume of a cone so we substitute the values of $katex is not defined$ and $katex is not defined$ into the formula.

$katex is not defined$
$katex is not defined$
$katex is not defined$
$katex is not defined$

katex is not defined

katex is not defined

katex is not defined

katex is not defined

Calculate the value of the radius by dividing the diameter by $katex is not defined$ , then substitute the values of $katex is not defined$ and $katex is not defined$ into the formula and solve.

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

Substitute the volume and the radius into the volume formula and then rearrange to find the perpendicular height, $katex is not defined$ .

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

Substitute the values of the volume and the perpendicular height into the formula, and rearrange to find the radius $katex is not defined$ .

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

katex is not defined

Volume of a cone GCSE questions

$katex is not defined$

1. Below is a cone with radius $katex is not defined$ and perpendicular height $katex is not defined$ .

Calculate the volume of the cone.

Give your answer to $katex is not defined$ significant figures.

(2 marks)

Show answer

katex is not defined

(1)

katex is not defined

(1)

2. Use the information in the diagram below to calculate the volume of the 3D shape.

Write your answer in terms of $katex is not defined$ .

(2 marks)

Show answer

katex is not defined

(1)

katex is not defined

(1)

3. A container is a cone of radius $katex is not defined$ and perpendicular height $katex is not defined$ .

Water fills the container at a rate of $katex is not defined$ per minute.

How long does it take to fill the container?

Give your answer to the nearest minute.

(4 marks)

Show answer

katex is not defined

(1)

katex is not defined

(1)

katex is not defined

(1)

$katex is not defined$ minutes

(1)

Learning checklist

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Calculate the volume of a cone

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