Geometry problem 4 by Dhaval Furia

The median of the equilateral triangle face $10\sqrt 3$ , half the width of the square base $10$ , and the height of the pyramid $h$ form a right triangle with the median of the equilateral triangle as the hypotenuse. By Pythagorean theorem ,

$\begin{aligned} h^2 & = (10\sqrt 3)^2 - 10^2 = 200 \\ \implies h & = \sqrt{200} = 10\sqrt 2 \end{aligned}$