Question

Equilateral hexagon is revolving around one of its edges. Find the volume of the solid of revolution.

no idea how to do this can someone please help!!! urgent!

Answer 1

One way: Theorem of Pappas
If the edge goes from (0,0) to (0,1), then the center is at (√3/2,√3/2).

The area of the hexagon is 3√3/2. The radius of rotation is √3/2, so its path has length π√3.

So, the volume of the solid is π√3 * 3√3/2 = 9π/2

Another way: calculus. Using symmetry, we can rotate a triangle and a rectangle about the y-axis.

Triangle: using discs, the volume is

v = ∫[1,3/2] π(R^2-r^2) dy
where R=(√3(3/2-y) and r=(√3(y-1/2))
v = ∫[1/2,1] π((√3(3/2-y))^2-(√3(y-1/2))^2) dy = 3π/4

Rectangle:
v = ∫[0,1/2] πr^2 dy
where r=√3
v = ∫[0,1/2] 3π dy = 3π/2

That is the volume of the top half of the figure: 9π/4

Double that and you get the first volume: 9π/2