A right prism has volume 3645 and bases that are regular hexagons with sides of length 9. Find the height of the prism.

Answer 1

area of base = (3 sqrt3) /2 * 9^2 = 210.4
so 210.4 h = 3645
h = 3645/210.4

Answer 2

The tricky part is to find the area of the hexagon
It is made of 6 equilateral triangles with sides 9 and angles of 60°
The area of one of them is (1/2)(9)(9)sin60
= 81/2(√3/2) = (81/4)√3
but we have 6 of them, so the area of the base = 486√3/4
then
486√3/4 h = 3645
243√3/2 h = 3645
h = 2(3645)/(243√3) = ....

Answer 3

[Formula for finding the volume of a prism]
Volume = Area of base * Height

Using algebra...
Height = Volume / Area of base

This is what we will use to find the height of the prism.

Given that the volume of the prism is 3645, all we need to find is the area of the base. The hexagonal base can be split into six congruent equilateral triangles. The side length of the base is 9; so, all we need to do is to this:

Area of base = 6 * ((9^2)*sqrt(3)) / 4 = 243 * sqrt(3) / 2

Here, I am using the formula of an equilateral triangle’s area: A = (s^2)*sqrt(3) / 4. In this problem, the side length is 9.

Now we have everything we need to find the height of the prism.

Height = Volume / Area of base = 3645 / (243 * sqrt(3) / 2) = 10 * sqrt(3)

Answer: The height of the prism is 10*sqrt(3) units.