The ball chapter and stereometry review

Euler’s formula: for a convex polyhedron V − E + F = 2. Prism V = S_base H, S_lat = P H (right); cylinder V = πr²H, S_lat = 2πrH; pyramid and cone V = ⅓ S_base H, with S_lat = πrl for the cone; ball V = (4/3)πR³, S = 4πR²; for frustums V = (H/3)(Q₁ + √(Q₁Q₂) + Q₂). For similar solids with linear ratio k, areas scale by k² and volumes by k³. In coordinates AB = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²), the midpoint of a segment has the averaged coordinates; the dot product of vectors is a · b = x₁x₂ + y₁y₂ + z₁z₂ = |a||b|cosφ, and a · b = 0 ⟺ a ⟂ b. We split a composite solid into simple parts and add the volumes, while for the surface we drop the internal (glued) faces. In entrance-exam tests a quick estimate of the answer (units, sign, rough size) catches mistakes.