Volume and surface area

Volume and surface area are the two fundamental measurements of a three-dimensional solid. Volume counts how much space the solid fills, measured in cubic units, while surface area totals the area of every face or curved surface that wraps the solid, measured in square units. A box that holds 24 cubic inches of sand and needs 52 square inches of wrapping paper has volume 24 in³ and surface area 52 in².

For prisms and cylinders, volume follows one pattern: base area × height. A rectangular box gives V = lwh; a cylinder gives V = πr²h. Pointed solids — pyramids and cones — hold exactly one-third of their prism counterparts, so a cone's volume is V = (1/3)πr²h. The sphere has V = (4/3)πr³ and surface area 4πr². Other key surface area results: a rectangular box has SA = 2lw + 2lh + 2wh, and a cylinder has SA = 2πr² + 2πrh.

Competition problems rarely stop at plugging into a formula. They combine solids (a sphere inscribed in a cube), track how measurements scale — doubling every length multiplies surface area by 4 and volume by 8 — or run comparisons, such as finding which of two containers holds more. Recognizing that similar solids scale areas by k² and volumes by k³ solves many problems in one line.

Volume and surface area questions appear regularly in the geometry portion of the AMC 10 and AMC 12, where knowing the formulas is assumed and the challenge comes from combining them with similarity, the Pythagorean theorem, and clever decompositions of composite solids.