# Why average area of the horizontal slices of the conical frustum doesn't work for it's volume?

I would like to react to one of the answers on this thread (I don't have enough rep to make a comment):

Use cylinder's formula for frustum (conical frustum)

Essentially, what you'd need is the average of the areas of the horizontal slices into which the frustrum is cut by planes paralell to its base, not their diameters.

I would like to ask if it is really working, I tried it with a conical frustum of r1 = 4, r2 = 2, h = 10 and a get result of V =100$\pi$ , the result give by formula $\frac1 3\pi h(r^2+R^2+rR)$ is $293.215$

Did I something wrong in the calculation or is the idea of "transforming" a cone frustum into cylinder with base area equal to an average area of two frustum's slices wrong ?

• I am sorry if I misunderstood something, but author of question in the link is asking why avg. of radii is not working, not the avg. of the areas. And the last answer in the link is states that he should take avg. of the areas instead of avg. of radii, but it does not work for me, so the matter of my question is whether I am wrong, or the answer in the link is wrong. Jan 16, 2016 at 20:23
• Oh, I see now, I've rescinded my close vote. Jan 16, 2016 at 20:53

$$\int_0^{h}\pi\left(r_1 - \frac{(r_1-r_2)}{h}r\right)^2 \;\mathrm{d}r.$$
This averages the area function where the radius changes linearly between $r_1$ and $r_2$ with $h$, and integrates to the formula you gave.