[justin_]

If you're looking for where the 1/3 "comes from", I think using Pappus' second theorem[0] is the way to go. 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 its _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!

The OP alludes to this with the mention of a solid of revolution in the post. And someone mentions Pappus in the thread there. How did Pappus figure out the centroid was distance to consider? I think that's another mystery, since the proofs of the theorem seem to depend on calculus, which brings us back to the original question.

Anyway, I was looking into the exact question of the OP a few months and this was the most satisfying answer I could find for where the 1/3 comes from geometrically.

[0] https://en.wikipedia.org/wiki/Pappus%27s_centroid_theorem#Th...

[sudosysgen]

The centroid makes sense intuitively when you think about it as a center of mass. Since it's the average position, for every portion that's father away there is, in a sense, a portion that's closer - the portion being equal either insofar as it is larger and closer to the center or smaller and father away, and since the size/volume of a ring increase linearly with distance in an intuitive manner, you can justify to yourself that you should be using the centroid without any formal calculus, just by considering the "weight" of a ring.

[justin_]

This is an excellent explanation. If we imagine the triangle being swept about in a circle, the points in the "outer" section (forming the circle at edge of the base our our cone) will cover more distance the points in the "inner" section (the points near the center line of our cone that we're revolving around). The distance moved by the centroid, then, is the average distance a point travels.

[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)