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Cones, spheres, pyramids and frustums

Use volume and surface area formulae for cones, spheres, pyramids and frustums, including similar-shape reasoning.

Geometry and MeasuresHigherGrades 6 to 9Skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > Curved solids and frustums

Pearson Edexcel GCSE Maths Higher geometry G16 and R12: use formulae for cones, spheres, pyramids and frustums, and apply similarity in 3D.

Practise this topicUpload your answerAsk AI to explain thisTry exam-style questions
Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

A pyramid has a polygon base and triangular faces meeting at a point. A cone has a circular base and one curved face.

Volume of a pyramid = 1/3 x base area x perpendicular height.

Volume of a cone = 1/3πr²h. The height must be perpendicular to the base, not the slant height.

Volume of a sphere = 4/3πr³. Surface area of a sphere = 4πr².

A frustum is made by cutting the top from a cone or pyramid. The reliable method is large solid minus small similar solid.

Similar solids use volume scale factor = linear scale factor³ and surface area scale factor = linear scale factor².

Higher note: if a frustum question gives two radii, use the radius ratio to connect the missing heights by similar triangles.

Key ruleCone and pyramid volumes have a 1/3. Frustum volume is large solid minus small similar solid.

Diagram guide

conespherepyramidHigher formulae need radius and heightcone and pyramid: 1/3 x base area x height
Higher solid formulaeCheck whether the question gives radius, perpendicular height, slant height, volume or surface area.
small conefrustumfrustum = large cone - small coneuse similar shapes to find missing height or radius
Frustum methodRebuild the missing small solid, then subtract it from the large solid.

Worked examples

Cone volume

A cone has radius 4 cm and perpendicular height 9 cm. Find its volume in terms of π.

conespherepyramidHigher formulae need radius and heightcone and pyramid: 1/3 x base area x height
Example: coneUse perpendicular height, not slant height.
  1. Volume = 1/3πr²h.
  2. Volume = 1/3 x π x 4² x 9.
  3. Volume = 48π cm³.

Answer: 48π cm³

Sphere volume

A sphere has radius 6 cm. Find its volume in terms of π.

conespherepyramidHigher formulae need radius and heightcone and pyramid: 1/3 x base area x height
Example: sphereSphere volume uses r³.
  1. Volume = 4/3πr³.
  2. Volume = 4/3 x π x 6³.
  3. Volume = 288π cm³.

Answer: 288π cm³

Frustum from similar cones

A large cone has radius 6 cm and height 12 cm. A similar small cone of radius 3 cm is removed from the top. Find the frustum volume in terms of π.

small conefrustumfrustum = large cone - small coneuse similar shapes to find missing height or radius
Example: subtract conesThe radius scale factor from large to small is 1/2, so the height is also 1/2.
  1. Small cone height = 12 x 1/2 = 6 cm.
  2. Large cone volume = 1/3 x π x 6² x 12 = 144π.
  3. Small cone volume = 1/3 x π x 3² x 6 = 18π.
  4. Frustum volume = 144π - 18π = 126π cm³.

Answer: 126π cm³

Common mistakes

  • Forgetting the 1/3 in cone or pyramid volume.
  • Using slant height instead of perpendicular height for volume.
  • Using diameter instead of radius.
  • Using surface area scale factor for volume questions.
  • Trying to use a frustum formula without understanding the similar solids.
  • Subtracting the small radius from the large radius instead of subtracting volumes.

Quick exercise

Try these before moving to the exam-style questions.

  1. A cone has radius 3 cm and height 10 cm. Find its volume in terms of π.
    conespherepyramidHigher formulae need radius and heightcone and pyramid: 1/3 x base area x height
    Quick check: cone volumeUse 1/3πr²h.
  2. A sphere has radius 2 cm. Find its surface area in terms of π.
    conespherepyramidHigher formulae need radius and heightcone and pyramid: 1/3 x base area x height
    Quick check: sphere areaSurface area of a sphere is 4πr².
  3. A pyramid has base area 45 cm² and perpendicular height 8 cm. Find its volume.
    conespherepyramidHigher formulae need radius and heightcone and pyramid: 1/3 x base area x height
    Quick check: pyramidUse 1/3 x base area x height.
  4. A linear scale factor is 3. What is the volume scale factor?
    small conefrustumfrustum = large cone - small coneuse similar shapes to find missing height or radius
    Quick check: similar solidsCube the linear scale factor for volume.
  5. For frustum volume, do you subtract lengths, areas or volumes?
    small conefrustumfrustum = large cone - small coneuse similar shapes to find missing height or radius
    Quick check: frustum methodFind the two solid volumes first.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleHigherCalculator3 marks

A cone has radius 5 cm and perpendicular height 12 cm. Work out its volume in terms of π.

conespherepyramidHigher formulae need radius and heightcone and pyramid: 1/3 x base area x height
Question diagram: coneUse the perpendicular height.
cone volumehigher measures3D solids
Standard exam styleHigherCalculator4 marks

A sphere has radius 7 cm. Work out its surface area to 3 significant figures.

conespherepyramidHigher formulae need radius and heightcone and pyramid: 1/3 x base area x height
Question diagram: sphere surface areaSurface area uses r², not r³.
sphere surface arearoundinghigher geometry
ChallengeHigherCalculator6 marks

A cone has radius 8 cm and height 15 cm. A similar smaller cone of radius 4 cm is removed from the top. Find the volume of the frustum in terms of π.

small conefrustumfrustum = large cone - small coneuse similar shapes to find missing height or radius
Question diagram: frustumUse similar cones, then subtract volumes.
frustumsimilar solidshigher geometry