Find the volume of a pyramid with height 8 and base an equilateral triangle with side $a = 8$.

Select the correct answer.
A) $\frac { 128 \sqrt { 3 } } { 3 }$
B) $128 \sqrt { 3 }$
C) $\frac { 3 \sqrt { 3 } } { 2 }$
D) $\frac { 128 \sqrt { 3 } } { 5 }$
E) $\frac { \sqrt { 3 } } { 10 }$

Question

Find the volume of a pyramid with height 8 and base an equilateral triangle with side $a = 8$.

Select the correct answer.
A) $\frac { 128 \sqrt { 3 } } { 3 }$ 
B) $128 \sqrt { 3 }$ 
C) $\frac { 3 \sqrt { 3 } } { 2 }$ 
D) $\frac { 128 \sqrt { 3 } } { 5 }$ 
E) $\frac { \sqrt { 3 } } { 10 }$

Quizplus · Accepted Answer

The area of an equilateral triangle with side $a$ is $A=\frac{\sqrt{3}}{4}a^2$. The height of the pyramid cuts the equilateral triangle into two 30-60-90 triangles, so the height of the triangle is $a\frac{\sqrt{3}}{2}$. The volume of the pyramid is $\frac{1}{3}Ah=\frac{1}{3}\cdot\frac{\sqrt{3}}{4}a^2\cdot8=\frac{2\sqrt{3}}{3}a^2$. Plugging $a=8$, we get the volume to be $\boxed{\frac{128\sqrt{3}}{3}}$.

Find the Volume of a Pyramid with Height 8 and Base