[Izkata]

> The theorem states that the volume of a solid of revolution formed by a revolving shape is the area of the original shape multiplied by the distance traveled by the _centroid_. The centroid of a triangle is determined based on distances from the three sides, and this is where thirds come from. For example, the centroid of a a right triangle with points (0,0), (0,1), and (1,0) sits at (0.33(3), 0.33(3)). That's a third!

Isn't the centroid of an equivalent rectangle rotated to make a cylinder at (0.5, 0.5)? 0.33 is not 1/3 of 0.5

[squirtlebonflow]

No, but it's 2/3. Then you divide by 2 because the area of the triangle is half the area of the rectangle.

[justin_]

To make an enclosing cylinder that bounds the cone in my example, we can use a 1x1 square with centroid (0.5, 0.5). The cylinder has volume π. And that works out from Pappus' theorem too:

1 (the square area) * 0.5*2π (the distance traveled by the centroid)