Right prism with rhombus base from its diagonals

Text

A right prism has a rhombus base whose diagonals are $8$ cm and $15$ cm; the prism height equals the rhombus side. Find the total surface area and the volume.

Solution

Rhombus side: $\ell=\sqrt{4^2+7.5^2}=\sqrt{16+56.25}=\sqrt{72.25}=8.5$ cm, so the height is $8.5$. Base area: $\dfrac{8\cdot15}{2}=60$; perimeter $=34$. Lateral surface: $34\cdot8.5=289$; total surface: $S_t=289+2\cdot60=409$ cm$^2$. Volume: $V=60\cdot8.5=510$ cm$^3$. $\boxed{S_t=409\ \text{cm}^2;\ V=510\ \text{cm}^3}$