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Similarity, congruence and scale
Lesson 49 of 75
Challenge

Frustums and similar solids

Objective. Use similarity to find missing dimensions and volumes.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
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Learn: the key idea

A frustum is the solid remaining when a smaller cone (or pyramid) is cut off the top of a larger one by a plane parallel to the base. To find its volume, calculate the volume of the full large cone and subtract the volume of the smaller cone that was removed, using similarity to find the smaller cone's dimensions if needed. If a cone of height 12cm is cut so the top removed piece has height 4cm, the removed cone is similar to the original with linear scale factor 4/12=1/3, so its volume is (1/3)³=1/27 of the original. For missing lengths, set up a ratio between corresponding dimensions of the small and large similar cones. Always confirm both cones share the same apex and parallel bases before applying similarity.

Worked example

A cone of height 9cm and volume 243cm³ has its top 3cm cut off to form a frustum, with the removed small cone similar to the original. Find the volume of the frustum.

Diagram of a cylinder
  1. 1Linear scale factor of removed cone to original: 3/9 = 1/3
  2. 2Volume scale factor = (1/3)³ = 1/27
  3. 3Volume of removed small cone = 243 × 1/27 = 9 cm³
  4. 4Frustum volume = 243 − 9 = 234 cm³

Answer: 234 cm³

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Common mistake. Pupils find a frustum's volume by only calculating the volume of the large cone, forgetting to subtract the smaller cone that was removed. This overestimates the frustum's volume substantially. Always subtract: frustum volume = large cone volume − small cone volume.

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