In a regular quadrangular pyramid, the apothem is 16, the lateral edge is 20. Find the height and side of the base.

Let’s denote the pyramid given by condition, SABCD, at the base – the square ABCD, SO – the height of the pyramid, point O – the center of the square, the point of intersection of the diagonals, SK – apothem. Consider a right-angled triangle SKA, find a leg AK – half of the side of the base.
AK = √ (SA² – SK²) = √ (400 – 256) = √144 = 12.
Base side:
AB = 2 * AK = 24.
Consider a right-angled triangle SOK and find a leg SO – the height of the pyramid.
SO = √ (SK² – OK²) = √ (256 – 144) = √112 = 4√7.
Answer: the side of the base is 24, the height of the pyramid is 4√7.