Cone surfaces

Turn a right triangle about one of its shorter sides and you get a right circular cone: height h, base radius r and slant height l, with l² = h² + r².

Unroll the curved surface: it is a sector of a circle of radius l, and its arc is the base circumference 2πr.

Area of that sector. Cut it into thin triangles of height l. Their bases add up to 2πr, so CSA = ½ × 2πr × l = πrl.

Another way: the sector's angle θ satisfies θ/360° = r/l, and its area is πl² × θ/360° = πrl. Same answer.

Total surface area adds the round base: πrl + πr² = πr(l + r).

Given the height, not the slant? Find l first. A cap with r = 7 and h = 24 has l = 25 (7-24-25 triangle), so CSA = (22/7) × 7 × 25 = 550 cm².