Question

The volume of a regular tetragonal pyramid is V and its altitude is H. Find the length of the apothem of the pyramid.

Answer 1

$b\to$ base length , $l\to$ slant height (apothem of pyramid)

$h\to$ altitude

Vol. of pyramid $=\frac{1}{3}$ (Area of base) $\times$ (Altitude)

$V=\frac{1}{3}\times \left(b^2\right)h\Rightarrow h=\frac{3V}{b^2}$

Now, $h^2+\frac{b^2}{4}=l^2$ (pythagoras th.)

$\Rightarrow \frac{{9V}^2}{b^4}+\frac{b^2}{4}=l^2\Rightarrow l=\sqrt{\frac{{9V}^2}{b^4}+\frac{b^2}{4}}$