Skip to content
VirtusAcademy

Volume of a Cone / Pyramid / Sphere

FoundationHigherAQAEdexcelOCR

Get to grips with volume of a cone / pyramid / sphere using these Foundation and Higher GCSE Maths practice questions. The worksheet focuses on volume of cones, pyramids and spheres, and the accompanying mark scheme breaks down each solution clearly. Suitable for AQA, Edexcel and OCR. Download the questions and answers for free. A cone or pyramid is one-third of the matching prism; a sphere is (4/3)πr³.

Free downloads

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.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Cones, pyramids and spheres have their own volume formulae, all given on the formula sheet at GCSE.

A pyramid's volume is \(\frac{1}{3} \times \text{base area} \times \text{height}\), and a cone follows the same pattern with a circular base, giving \(\frac{1}{3}\pi r^2 h\). A sphere's volume is \(\frac{4}{3}\pi r^3\).

The recurring error is forgetting the fraction. A cone is exactly one third of the cylinder that encloses it, so leaving out the \(\frac{1}{3}\) triples the answer. Writing the formula out before substituting prevents this.

Revision notes

Pyramids and cones

Volume is \(\frac{1}{3} \times \text{base area} \times \text{perpendicular height}\).

For a cone the base is a circle, giving \(\frac{1}{3}\pi r^2 h\). A cone of radius \(3\)cm and height \(6\)cm has volume \(\frac{1}{3} \times \pi \times 9 \times 6 = 18\pi\)cm³.

Spheres

A sphere's volume is \(\frac{4}{3}\pi r^3\), where the radius is cubed.

For radius \(3\)cm: \(\frac{4}{3} \times \pi \times 27 = 36\pi\)cm³. A hemisphere is half of this.

Perpendicular height

For a cone or pyramid, the height is measured from the apex straight down to the base, not along the slanted edge.

The slant height is a different measurement, used for surface area rather than volume.

Key points

  • Pyramid volume is \(\frac{1}{3} \times\) base area \(\times\) height.
  • Cone volume is \(\frac{1}{3}\pi r^2 h\).
  • Sphere volume is \(\frac{4}{3}\pi r^3\).
  • Never forget the fraction.
  • Use the perpendicular height, not the slant.
  • A hemisphere is half a sphere.

Worked examples

Example 1

Find the volume of a cone with radius \(6\)cm and height \(10\)cm, in terms of \(\pi\).

Working

\[\frac{1}{3}\pi r^2 h = \frac{1}{3}\pi \times 36 \times 10\]substitute into the formula
\[= 120\pi\text{cm}^3\]evaluate the volume

Example 2

Find the volume of a sphere with radius \(3\)cm, in terms of \(\pi\).

Working

\[\frac{4}{3}\pi r^3 = \frac{4}{3}\pi \times 27\]cube the radius first
\[= 36\pi\text{cm}^3\]evaluate the volume

Example 3

Find the volume of a pyramid with base area \(24\)cm² and height \(9\)cm.

Working

\[\frac{1}{3} \times 24 \times 9\]use the pyramid formula
\[= 72\text{cm}^3\]state the volume

Common mistakes

  • Forgetting the \(\frac{1}{3}\) for a cone or pyramid.

    This triples the answer. A cone is a third of the enclosing cylinder.

  • Using the slant height for volume.

    Volume needs the perpendicular height from apex to base.

  • Cubing the wrong quantity for a sphere.

    It is the radius that is cubed, not the whole expression.

  • Using the diameter instead of the radius.

    All three formulae use the radius.

Exam tips

  • Write the formula out fully before substituting.
  • Check whether the height given is perpendicular or slant.
  • Cube the radius as a separate step for spheres.
  • Leave answers in terms of π when exact values are wanted.

Key terms

Cone
A solid with a circular base tapering to a point.
Sphere
A perfectly round solid.
Apex
The point at the top of a cone or pyramid.
Slant height
The distance along the sloping surface, used for surface area.

Related topics

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