Square pyramid from edge and face apothem

Text

A regular square pyramid has base edge $12$ cm and lateral-face slant height (apothem) $10$ cm. Find the height, the total surface area and the volume.

Solution

The base apothem is $6$; the height is $h=\sqrt{10^2-6^2}=8$ cm. Lateral surface: $\dfrac12\cdot(4\cdot12)\cdot10=240$; base area $=144$. Total surface: $S_t=240+144=384$ cm$^2$. Volume: $V=\dfrac13\cdot144\cdot8=384$ cm$^3$. $\boxed{h=8\ \text{cm};\ S_t=384\ \text{cm}^2;\ V=384\ \text{cm}^3}$