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Cone: from slant height to perpendicular height

2026-09-09Spatial GeometryMs Tam

ℓ, r and h meet in a right triangle: ℓ² = r² + h².

Key formulas

ℓ² = r² + h²
V = (1/3)πr²h

Cut the cone

A vertical cross-section through the apex is an isosceles triangle. Half of it is right-angled with legs r, h and hypotenuse ℓ.

Example

r=5, ℓ=13 → h=√(169−25)=12. Then V=(1/3)π(25)(12)=100π.

Surface note

Curved surface uses πrℓ, which needs the slant, not h. Volume needs h, not ℓ. Keep the roles separate.

Worked example

Worked example

Problem. A cone has r = 9 cm and slant height ℓ = 15 cm. Find h and the volume in terms of π.

Solution. h = √(15² − 9²) = √(225 − 81) = √144 = 12. V = (1/3)π × 81 × 12 = 324π cm³.

3-4-5

r=3, h=4, ℓ=5 is a classic quick check.

Review quiz

Cone r=8, ℓ=17. Height is:

Practice questions

  1. r=9, ℓ=15. Find h and V in terms of π.
  2. Why can ℓ never be shorter than r?

Study notes

Is slant height on the net?

Yes — the sector radius of the net equals ℓ.

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