Surface area of a sphere

What the symbols mean

$S$

the surface area

$r$

the radius

When to use it

Use it for the leather on a football, the paint on a dome, the area of a planet's surface, or the heat a spherical tank loses. A hemisphere with its flat base has $2\pi r^2+\pi r^2=3\pi r^2$.

Why it works

Archimedes again: the sphere's surface equals the curved surface of the cylinder that wraps it, $2\pi r\times 2r=4\pi r^2$. The derivative of the volume $\frac{43}{\pi }{r}^3$ with respect to r is also $4\pi r^2$: adding a thin skin adds surface area times thickness.

Common mistakes

• Using $\pi r^2$, which is only the area of a flat circle through the centre.
• Cubing r: surface area is in square units, so r is squared.