Pyramid volume from total surface and ratio

Text

A regular square pyramid has total surface area $225$ cm$^2$ and its base edge is $\dfrac83$ of its height. Find the base edge and the volume.

Solution

Let $a$ be the base edge and $h=\dfrac38 a$ the height. The base apothem is $\dfrac a2$; the face apothem is $\sqrt{h^2+\left(\tfrac a2\right)^2}=\sqrt{\tfrac{9}{64}a^2+\tfrac14a^2}=\sqrt{\tfrac{25}{64}a^2}=\tfrac58 a.$ Lateral surface $=\dfrac12(4a)\cdot\dfrac58a=\dfrac54a^2$; total surface $=a^2+\dfrac54a^2=\dfrac94a^2=225$: $a^2=100\Rightarrow a=10,\quad h=\tfrac38\cdot10=3.75.$ Volume: $V=\dfrac13\cdot100\cdot3.75=125$ cm$^3$. $\boxed{a=10\ \text{cm};\ V=125\ \text{cm}^3}$