# 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 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?

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 $d = 4\sqrt{2}$

Hint:

Create a triangle where the hypotenuse is a line in one of the triangles (thus the height of one of the triangles), and then the two other sides are of length half the base and the height of the pyramid. Solve for this last side using pythagorean theorem.

Let the pyramid have top X, and base ABCD. Then triangle XBD and ABC are congruence (SSS) (using the fact that the base is square and the side is equilateral). Height of pyramid is easily shown to be height of XBD. So you just need to find height of ABC from A to BC, which is simply $2\sqrt{2}$.

Hint: First find $|AB|$ and then focus on finding the height of pyramid in red triangle.

Use the picture in Mathlover's answer for visual help.

• $A=4\cdot4=16$

Find $|AB|$. Use Pythagorean Theorem.

• $\displaystyle \left(\frac{4}{2} \right)^2+(|AB|)^2=(4)^2 \implies |AB|=2\sqrt{3}$

Find the height $h$ of the pyramid. (This is the $\color{red}{\text{red}}$ triangle in their picture.)

• $\displaystyle \left(\frac{2}{2} \right)^2+h^2=(2\sqrt{3})^2 \implies h=2\sqrt{2}$

Now we need to find the base area $A$. Since the sides of the pyramid are equilateral triangles, so is the base side. We have $$A=\frac{1}{3}Ah=\frac{1}{3} (16)(2\sqrt{2})=\frac{32\sqrt{2}}{3}\approx\boxed{15.08}$$ I like choice (E).