Surface area and volume

• Cuboid: V = lwh, surface area = 2(lw + lh + wh).
• Prism (any solid with uniform cross-section): V = Al. Cylinder: V = πr²h, curved surface area = 2πrh.
• Pyramid: V = ⅓Ah. Cone: V = ⅓πr²h, curved surface area = πrl, where l is the slant height.
• Sphere: V = ⁴⁄₃πr³, surface area = 4πr².
• The List of formulas gives all the volumes and curved surface areas above; the cuboid surface area and the flat faces you work out.
• Answers may be asked for in terms of π.

Three pairs of rectangles

A cuboid has three pairs of equal rectangular faces.

Cross-section × length

A prism has the same cross-section all the way along. This includes a cylindrical sector, a solid whose cross-section is a sector of a circle. Surface area = 2 × cross-section + (perimeter of cross-section × length).

A circular prism

A closed cylinder has a curved surface plus two circular ends. Unrolled, the curved surface is a rectangle 2πr long and h wide.

One third of the prism

A pyramid or cone has one third of the volume of the prism or cylinder with the same base and height. The cone formula uses the vertical height h for volume and the slant height l for the curved surface: l² = r² + h².

Volume and surface area