Frustum of a cone

Properties

• 2 flat circular faces of different radii and 1 curved face.
• The slant height, the height and the difference of the radii form a right-angled triangle.
• When the two radii are equal, the formulas become those of a cylinder. When the smaller radius is 0, they become those of a cone.

Why the formulas work

A frustum is a big cone with a small cone removed from its top. The two cones are similar, and subtracting their volumes and tidying the algebra gives ⅓πh(R² + Rr + r²). The curved surface is found the same way, by subtracting the curved surfaces of the two cones, and comes to π(R + r)l.

Worked example

Take a frustum of a cone with larger radius 14 cm, smaller radius 7 cm and height 24 cm.

1. Slant height. Slant height l = √(h² + (R − r)²). Here: √(24² + (14 − 7)²) = 25 cm.
2. Volume. Volume = ⅓ πh(R² + Rr + r²). Here: ⅓ × π × 24 × (14² + 14 × 7 + 7²) = 8,620.53 cm³ (with π = 22/7: 8,624 cm³).
3. Curved surface area. Curved surface area = π(R + r)l. Here: π × (14 + 7) × 25 = 1,649.34 cm² (with π = 22/7: 1,650 cm²).
4. Total surface area. Total surface area = π(R + r)l + π(R² + r²). Here: 1,649.34 + π × (14² + 7²) = 2,419.03 cm² (with π = 22/7: 2,420 cm²).

Finding a missing measurement

If the slant height is given and not the height, h = √(l² − (R − r)²): radii of 14 cm and 7 cm with a slant height of 25 cm give h = √(625 − 49) = 24 cm. For a bucket, the capacity in litres is the volume in cm³ divided by 1000.

Mistakes to avoid

• Using the vertical height in the curved surface formula. It needs the slant height.
• For an open bucket, add only the base circle, not both.
• Averaging the radii and using the cylinder formula gives a wrong volume.

Where you see it

Buckets, drinking glasses, flower pots, lampshades and funnels (the wide part) are frustums.

Try these

Work each one out, then tap to check.

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