Cross sections to get a radius

Question

The base of a solid is a circle of a certain radius. The cross-sections to a diameter of the base are equilateral triangles. The solid has volume 10 cubic meters.

I need to find the radius but every method I use comes up with a different answer.

Answer 1

Let the base of the solid be the circle x²+y²=r²

Consider the stack of triangles making up the solid. The triangle at distance x from the center of the circle has base 2y = 2√(r²-x²).

You know that the area of an equilateral triangle of side s is √3/4 s²

So, each triangle at distance x from the center has area √3/4 (2√(r²-x²))

For the volume of the solid, we need to consider the stack of triangles, each of thickness dx. Due to the symmetry of the region, we can now see that the volume is

v = 2∫[0..r] √3/4 (2√(r²-x²)) dx = 2/√3 r³

Since we know that the solid has volume 10 cm³, we just need to solve

2/√3 r³ = 10

r³ = 5√3

r = ³√(5√3)