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 $=20cm$

Area of hexagon $=6\times \frac{\sqrt{3}}{4}\times {\left(2a\right)}^2=6\sqrt{3}{a}^2$

Slant height of pyramid $=\frac{59}{2}cm$

$HF=\frac{5a}{2}$ (slant height)

$OH=$ Height $\left(h\right)$

Height $=\sqrt{{\left(\frac{{5a}^2}{2}\right)}^2-{\left(2a\right)}^2}=\sqrt{\frac{25}{4}{a}^2-4a^2}=\frac{3a}{2}$

$\therefore$ Volume of pyramid $=\frac{1}{3}ar.ofbase\times height$

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

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