Pyramid with a lateral edge perpendicular to the base

Text

A pyramid has a square base of edge $6$ cm; one lateral edge is perpendicular to the base at a vertex and measures $8$ cm. Find the volume and the total surface area.

Solution

The perpendicular edge is the height $h=8$. Volume: $V=\dfrac13\cdot36\cdot8=96$ cm$^3$. Two lateral faces are right triangles with legs $6$ and $8$: area $\dfrac12\cdot6\cdot8=24$ each. The other two faces have height $\sqrt{6^2+8^2}=10$: area $\dfrac12\cdot6\cdot10=30$ each. Total surface: $S_t=36+2\cdot24+2\cdot30=144$ cm$^2$. $\boxed{V=96\ \text{cm}^3;\ S_t=144\ \text{cm}^2}$