The cone, the frustum and chapter review

Rotating a right triangle about one of its legs gives a cone: that leg is the axis and height h, the other leg is the base radius r and the hypotenuse is the slant height l, so l² = r² + h². The axial section is an isosceles triangle with base 2r and sides l. Unrolling the lateral surface gives a circular sector with radius l and arc length 2πr; its central angle is 360° · r/l and its area is ½ · l · 2πr, so S_lat = πrl and S_total = πr(r + l). The volume is V = ⅓ S_base h = ⅓ πr²h (obtained as the limit between inscribed and circumscribed pyramids). A plane parallel to the base cuts a similar cone and leaves a frustum; for the frustum S_lat = πl(r₁ + r₂) and V = (πH/3)(r₁² + r₁r₂ + r₂²), where l² = H² + (r₁ − r₂)². For similar cones the areas scale by k² and the volumes by k³.