Question

A triangular pyramid has a height of `x`, a side length of `2sqrt3`, and a volume of `7sqrt3`. What are the steps in order to find the height of the pyramid?

Answer 1

The height of the pyramid is 7

Explanation:

1) The formula of the pyramid's volume is `V = (S_(base) * h)/3`

2) Since the base is a equilateral triangle, `tan(60^o)=(height_(triangle))/((side)/2)`
Then `height_(triangle) = (sqrt(3)/2)*side`

3) The `Area_(triangle) = S_(base) = (base*height_(triangle))/2`
(Base of a equilateral triangle is equal to side, since all the sides are equal)

From (2) and (3) we get
`S_(base) = (side*(sqrt(3)/2)*side)/2 = (sqrt(3)/4)*side^2`

From (1) and the previous formula of the triangle's area, we get
`V = ((sqrt(3)/4)*side^2*h_(pyramid))/3 = (sqrt(3)/12)*side^2*h_(pyramid)`
So `h_(pyramid)=(12/sqrt(3))*V/(side^2)`

Now we only need to input the known values
`h_(pyramid)=(12/cancel(sqrt(3)))*(7*cancel(sqrt(3)))/(2*sqrt(3))^2 = (12*7)/(4*3) = 7`