Frustum volume
Objective. Use subtraction and similarity to find frustum measures.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A frustum is the solid left when a smaller cone or pyramid is sliced off the top of a larger one, parallel to its base. Its volume is found by calculating the volume of the full large cone/pyramid and subtracting the volume of the removed small one. Similar solids give the small cone's height and radius proportional to the large one, found using the ratio of corresponding lengths. For a cone of height 12 and radius 6 with the top 4 cm (by height) cut off, the small cone is similar with scale factor 4/12, so its radius is 2. Subtract volumes carefully using the cone formula ⅓πr²h for each.
Worked example
A cone of height 9 cm and base radius 6 cm has its top 3 cm (in height) cut off to leave a frustum. Find the radius of the cut, smaller cone's base.
- 1Small cone height is 3 cm out of total 9 cm
- 2Scale factor = 3/9 = 1/3
- 3Small cone radius = 6 × 1/3
- 4Radius = 2 cm
Answer: 2 cm
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Practice
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Start practisingCommon mistake. Students often forget that the removed cone is similar to the whole cone and just guess its radius instead of scaling it correctly, which gives a badly wrong subtracted volume.
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