Question

Find the volume of a regular tetragonal pyramid if a side of its base is "a" long and the base dihedral angle is $\alpha$.

Answer 1

Side of the base $=a$

Base dihedral angle $=\alpha$

Attitude $=?$

$\Rightarrow \tan \alpha =\frac{x}{a}$

$\Rightarrow x=a\tan \alpha$

$\Rightarrow$ Attitude $=a\tan \alpha$

$\Rightarrow$ Volume $=\frac{2}{3}(a^2-x^2)x$

$=\frac{2}{3}(a^2-a^2{\tan }^2\alpha )a\tan \alpha$

$=\frac{2}{3}{a}^3(1-{\tan }^2\alpha )\tan \alpha .$

Hence, the answer is $\frac{2}{3}{a}^3(1-{\tan }^2\alpha )\tan \alpha .$