Surface Area and Volume of Solids

Surface area and volume answer two different questions about the same solid. Surface area asks how much material covers the outside; volume asks how much space sits inside. A cylinder makes the connection visible: slice off the top and bottom circles, then unroll the tube, and you get a flat rectangle. The rectangle's width is the circumference of the circle, 2 π r, and its height is the cylinder's height, h. Add the two circles, π r² each, and the total surface area is 2 π r² + 2 π r h. Volume works differently: it is just the area of the circular base, π r², times how far that base extends, h, giving π r² h.

Cones and spheres follow the same spirit but with their own formulas: a cone's volume is (1/3) π r² h, and a sphere's is (4/3) π r^3. Given a shared radius r and a height of 2r, a cone holds (2/3) π r³, a sphere holds (4/3) π r³, and a cylinder holds 2 π r³ — a clean 1 to 2 to 3 ratio worth memorizing. This is also why a tall thin can and a short wide can built from equal amounts of metal hold different amounts: surface area depends on r and h in one combination, volume in another, so matching the skin doesn't match the space inside.

These solids sit next to area and perimeter work on flat shapes — the same base formulas for circles and triangles reappear here, just stretched into three dimensions. Expect to combine a volume formula with simple algebra, like solving for an unknown height once the volume is given.