Regular square pyramid from edge-height sum

Text

In a regular square pyramid the sum of the base edge and the height is $35$ cm and their ratio (edge : height) is $3:2$. Find the lateral surface and the volume.

Solution

With ratio $3:2$ and sum $35$: base edge $=21$ cm, height $h=14$ cm. Base apothem (half the edge) $=10.5$; slant height of a lateral face: $a=\sqrt{h^2+10.5^2}=\sqrt{196+110.25}=\sqrt{306.25}=17.5.$ Lateral surface: $S_\ell=\dfrac12\cdot(4\cdot21)\cdot17.5=\dfrac12\cdot84\cdot17.5=735$ cm$^2$. Volume: $V=\dfrac13\cdot21^2\cdot14=\dfrac13\cdot441\cdot14=2058$ cm$^3$. $\boxed{S_\ell=735\ \text{cm}^2;\ V=2058\ \text{cm}^3}$