Question

A solid has a circular base of radius 1. It has parallel cross-sections perpendicular to the base which are equilateral triangles. How do you find the volume of the solid?

Answer 1

`V = (4sqrt3)/3`

Explanation:

Set the origin of the axis at the center of the circle so that the origin is in the center and the parallel cross-sections have the base of the triangles parallel to the `y` axis.

For every `x in (-1,1)` the length of the base of the triangle will than be:

`b = 2sqrt(1-x^2)`

and since it is equilateral its height will be:

`h=sqrt(3)/2b = sqrt(3)sqrt(1-x^2)`

The area of the triangle is then:

`S = (bh)/2 = sqrt(3)(1-x^2)`

and the relative volume element:

`dV = sqrt(3)(1-x^2)dx`

Integrating over the interval:

`V = int_(-1)^1 sqrt(3)(1-x^2)dx = sqrt(3) [x-x^3/3]_(-1)^1`

`V = sqrt(3) (1-1/3+1-1/3) =(4sqrt(3))/3`