The ball chapter and stereometry review

Chapter formulas: ball V = (4/3)πR³, sphere S = 4πR², segment V = πh²(3R − h)/3, cap area 2πRh, sector V = ⅔πR²h. For a ball inscribed in a cube of edge a, r = a/2, and for a ball circumscribed about it, R = a√3/2. A cylinder circumscribed about a ball has height H = 2R and base radius R; then V_ball : V_cylinder = 2 : 3 and the surfaces (S_ball : S_total of the cylinder) are also 2 : 3 — Archimedes’ relation. For a cone (base radius r, height h, slant height l), the inscribed ball has radius ρ = rh/(r + l) (the incircle of the axial triangle) and the circumscribed ball has radius R = l²/(2h). If a polyhedron is inscribed in a ball, the centre is equidistant from all its vertices; if it is circumscribed about a ball, the centre is equidistant from all its faces. When solving, draw the axial section: in most cases the problem reduces to a plane problem about circles and triangles.