Surface Area Formulas

Surface area is the sum of the areas of every face of a solid. Sphere: 4πr². Cylinder: 2πr² (two ends) + 2πrh (the wrapped lateral face). Cone: πr² + πrl where l = √(h² + r²) is the slant height. Cube: 6s². Rectangular prism: 2lw + 2lh + 2wh.

Archimedes' Beautiful Result

The sphere's surface area, 4πr², exactly matches the lateral area of its circumscribed cylinder — a result so important Archimedes asked for it engraved on his tomb. It also explains why the sphere is the most efficient shape: the least surface for a given volume.

Why Surface Area Matters

Heat loss, chemical reaction rates, paint coverage, and packaging cost all scale with surface area. The surface-area-to-volume ratio is fundamental in biology and engineering: smaller objects have proportionally more surface, which is why a cup of tea cools faster than a large pot.