A pyramid has a square base with sides of length 4. If the sides of the pyramid are equilateral triangles, what is the pyramid's volume?
(A) 9.66
(B) 11.39
(C) 12.58
(D) 14.12
(E) 15.08
I know that the formula for the volume of a pyramid is base times height, but how do we find the height? Would it just be the altitude of one of the equilateral triangles?
Best Answer
Its E. Key step is: $h$ = height, then: $h^2 = 4^2 - \left(\dfrac{d}{2}\right)^2 = 16 - \left(\dfrac{4\sqrt{2}}{2}\right)^2 = 8$. So: $h = 2\sqrt{2}$, and therefore:
$V = \dfrac{Sh}{3} = \dfrac{4^2\cdot 2\sqrt{2}}{3} \approx 15.08$, $d$ = diagonal of the square at base, and $d = 4\sqrt{2}$