Surface Area

Surface area is the total area of all the outer faces of a three-dimensional solid — what you would need to wrap or paint. Units are squared (cm², m²), unlike volume which is cubed.

Common formulas:

• Cube: $SA = 6s^2$
• Rectangular prism: $SA = 2(lw + lh + wh)$
• Cylinder (closed): $SA = 2\pi r^2 + 2\pi r h$
• Sphere: $SA = 4\pi r^2$
• Cone (closed): $SA = \pi r^2 + \pi r \ell$ where $\ell = \sqrt{r^2 + h^2}$ is the slant height.

The surface-area-to-volume ratio is critical in biology (cells absorb nutrients through surface, store volume — limits cell size), engineering (heat dissipation), and chemistry (reaction rates). The sphere has the smallest surface area for a given volume, which is why bubbles and stars are spherical.