Cone volume when slant height is given

Recover the perpendicular height from radius and slant height before using (1/3)πr²h.

Key formulas

The volume formula needs the perpendicular height. If the question gives slant height l, first form a right triangle with r and h, then find h.

A cone has radius 5 cm and slant height 13 cm. Then h² = 13² − 5² = 169 − 25 = 144, so h = 12 cm. Volume = (1/3)π × 25 × 12 = 100π cm³.

Because 5-12-13 is a Pythagorean triple, the arithmetic is tidy. If h came out larger than l, the subtraction was reversed.

Problem. A cone has radius 5 cm and slant height 13 cm. Find its volume in terms of π.

Solution. h = √(ℓ² − r²) = √(169 − 25) = √144 = 12 cm. V = (1/3)πr²h = (1/3)π × 25 × 12 = 100π cm³.

r = 8, l = 17 → h = 15, since 8² + 15² = 64 + 225 = 289 = 17².

Cone r=6, slant height 10. Volume is:

Yes. It is the distance from the apex to a point on the base circumference along the side.

Solve V = (1/3)πr²h for h.