Question

There is a pyramid on a base which is a regular hexagon of side 2a. If every slant edge of this pyramid is of length $\frac{\left(5a\right)}{2}$, then the volume of this pyramid must be

• A. $3a^3$
• B. $3a^3\sqrt{2}$
• C. $3a^3\sqrt{3}$
• D. $6a^3$

Answer 1

The correct option is C $3a^3\sqrt{3}$

Side of regular hexagon $=2a$ $cm$

$\therefore$ ar. of hexagon $=6\times \frac{\sqrt{3}}{4}\times 2a\times 2a=6\sqrt{3}{a}^2$

Slant height of pyramid $=\frac{5a}{2}cm=AG$

$\therefore h=\sqrt{{\left(\frac{5a}{2}\right)}^2-{\left(2a\right)}^2}=\frac{3a}{2}$

$\therefore$ Volume of pyramid $=\frac{1}{3}\times ar.base\times h$

$=\frac{1}{3}\times 6\sqrt{3}{a}^2\times \frac{3a}{2}$

$=3\sqrt{3}{a}^3{cm}^3$