Surface Area of Pyramids & Cones

The formulas

For a regular pyramid, total surface area = base area + $\dfrac{1}{2} P \ell$, where $P$ is the base perimeter and $\ell$ is the slant height. The $\dfrac{1}{2} P \ell$ piece is just the triangular faces added up.

For a cone, total surface area = $\pi r^2 + \pi r \ell$. The $\pi r^2$ is the circular base and $\pi r \ell$ is the curved lateral surface, again using slant height $\ell$.

Slant height versus vertical height

The vertical height $h$ runs straight up from the center of the base to the apex. The slant height $\ell$ runs up the middle of a face. They form a right triangle with half the base, so $\ell = \sqrt{h^2 + (\text{half base})^2}$.

The figure is that right triangle: the vertical height $h$ and the base radius $r$ (or half the base edge for a pyramid) are the legs, and the slant height $\ell$ is the hypotenuse.

If a problem gives you $h$ but the formula needs $\ell$, use that right triangle to find $\ell$ first.

Worked examples

Try one yourself

Common questions