Right pyramid with a square base

Statement

A right pyramid has a square base with side $6$ and height $4$. Calculate its volume, the slant height and the lateral surface area.

Solution

Volume: $V = \tfrac{1}{3}S_b\cdot h = \tfrac{1}{3}\cdot 36\cdot 4 = \boxed{48}.$ The apothem of the base is half the side: $m = 3$. Slant height of the pyramid: $a = \sqrt{m^2+h^2} = \sqrt{9+16} = \boxed{5}.$ Lateral surface area (base perimeter $p_b = 24$): $S_{\text{lat}} = \tfrac{1}{2}\, p_b\cdot a = \tfrac{1}{2}\cdot 24\cdot 5 = \boxed{60}.$

Links

Topics: Synthetic geometry in space
Concepts: Apothem · Pyramid · Surface area · Volume
Skills: Calculating · Using formulae
Exercise types: Geometric problem