Mensuration - Right Square Pyramid

What is a Right Square Pyramid?

Right Square Pyramid has an square as its base. Rest of its faces are congruent (exactly same) isosceles or equilateral triangles. So, its apex is directly above the center of its base.

Formulae related to Right Square Pyramid

Formula 1: Slant Height and Slant Edge

If h - height, a - side of base square, l - slant height

Inradius of the square as the base, r = $\frac{a}{2}$
Circumradius of the square as the base, R = $\frac{a}{\sqrt{2}}$

By Pythagoras theorem:

Slant height, l = $\sqrt{h^2 + r^2}$ = $\sqrt{h^2 + (\frac{a}{2})^2}$

Slant Edge = $\sqrt{h^2 + R^2}$ = $\sqrt{h^2 + (\frac{a}{\sqrt{2}})^2}$

Formula 2: Volume

Volume of a Pyramid = $\frac{1}{3}$ × Base Area × Height

So, Volume of a Right Square Pyramid = $\frac{1}{3}$ × Base Area × Height = $\frac{1}{3}$ × $a^2$ × h

Formula 3: Surface Area

Lateral surface area = $\frac{1}{2}$ × Perimeter of base × Slant height = $\frac{1}{2}$ × 4a × l

Total surface area = Lateral surface area + Area of base = $\frac{1}{2}$ × 4a × l + $a^2$