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Volume of a Frustum

HigherHigher tier onlyAQAEdexcelOCR

This free Higher GCSE Maths worksheet on volume of a frustum helps you revise calculating the volume of a frustum. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA, Edexcel and OCR. Find the whole cone's volume and subtract the small cone removed from the top.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A frustum is what remains when the top of a cone or pyramid is cut off parallel to the base. It looks like a cone with its point removed.

The volume is found by subtraction: work out the volume of the complete cone, then subtract the volume of the small cone that was removed.

The two cones are mathematically similar, which is what makes the small one's dimensions findable. If the small cone's height is half the original's, its radius is also half, and its volume is one eighth — since volume scales with the cube of the linear scale factor.

Revision notes

The subtraction method

Volume of frustum equals volume of the whole cone minus volume of the removed top.

Both use \(\frac{1}{3}\pi r^2 h\), so calculate each separately and subtract.

Finding the small cone's dimensions

The two cones are similar, so their dimensions are in the same ratio.

If the frustum's top radius is \(2\)cm and the base radius is \(6\)cm, the small cone is one third the size, so its height is one third of the full cone's height.

Using the scale factor

Volume scales with the cube of the linear scale factor.

So if the small cone is half the height, its volume is \(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\) of the whole, and the frustum is the remaining \(\frac{7}{8}\).

Key points

  • A frustum is a cone or pyramid with the top cut off.
  • Volume is whole cone minus removed top.
  • The two cones are mathematically similar.
  • Dimensions are in the same ratio.
  • Volume scales with the cube of the scale factor.
  • Both volumes use \(\frac{1}{3}\pi r^2 h\).

Worked examples

Example 1

A full cone has volume \(96\pi\)cm³ and the removed top has volume \(12\pi\)cm³. Find the frustum's volume.

Working

\[96\pi - 12\pi\]subtract the removed top from the whole
\[= 84\pi\text{cm}^3\]state the frustum's volume

Example 2

A small cone is half the height of the full cone. What fraction of the volume does it have?

Working

\[\left(\frac{1}{2}\right)^3\]volume scales with the cube of the scale factor
\[= \frac{1}{8}\]state the fraction

Example 3

Find the volume of a frustum from a cone of radius \(6\)cm and height \(12\)cm, with the top \(6\)cm removed.

Working

\[\frac{1}{3}\pi \times 36 \times 12 = 144\pi\]find the whole cone's volume
\[\frac{1}{3}\pi \times 9 \times 6 = 18\pi\]the small cone has half the dimensions
\[144\pi - 18\pi = 126\pi\text{cm}^3\]subtract to find the frustum

Common mistakes

  • Subtracting the heights rather than the volumes.

    You must calculate both volumes and then subtract.

  • Assuming the small cone's radius equals the frustum's top radius without checking similarity.

    The cones are similar, so use the ratio of the dimensions.

  • Using the linear scale factor for volume.

    Volume scales with the cube of the scale factor, not the scale factor itself.

  • Forgetting the \(\frac{1}{3}\) in either cone volume.

    Both cones need the fraction.

Exam tips

  • Calculate both cone volumes separately before subtracting.
  • Use similarity to find the small cone's dimensions.
  • Remember volume scales with the cube of the scale factor.
  • Leave answers in terms of π when exact values are wanted.

Key terms

Frustum
A cone or pyramid with the top cut off parallel to the base.
Similar
Having the same shape with dimensions in a fixed ratio.
Scale factor
The ratio between corresponding lengths.
Apex
The point removed when forming a frustum.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.