What is the midsection of a frustum and how do you calculate its perimeter?

Question

A doubt while reading "How to solve it" by George Polya.

Given the figure below:

What is the midsection of this figure and how would you calculate its perimeter? (Would be great if you could tell me how to find it on the diagram).

Quoting from the book the midsection is defined as :

We call here mid-section the intersection of the frustum with a plane which is parallel both to the lower base and to the upper base of the frustum and bisects the altitude.

So is it the dotted area or the area with the solid line in the figure? You see this does not make since to me because he says the midsection bisects the altitude. There is no such construct on the given figure, so it must be something else. Also how can a plane bisect the altitude, a planar figure is 2D right? And the height is a 3D aspect right?

How would you also calculate the perimeter of that midsection ?

If can please do briefly describe what a midsection is? Is it something that only exists in solid objects or is present in objects of planar geometry as well?

Answer 1

Just $$2\pi\cdot\frac{R+r}{2}=\pi(R+r).$$

Because a perimeter of the circle with radius $x$ it's $2\pi x$.

The needed midsection it's a circle with diameter, which is a midline of the trapezoid with bases $2R$ and $2r$ and this midline is equal to $\frac{2R+2r}{2}=R+r.$

Id est, the radius of the circle is equal to $\frac{R+r}{2}.$

Answer 2

The midsection as defined in the question is the disk colored in yellow here:

It is necessary to appreciate that there is a proportionality between radius and height from apex by means of similar triangles. The straight generator of cone passes through origin at apex of originating cone. You can find radius and height even perimeter at half, quarter, three quarter frustum height also.

Also let $x$ be distance from cone apex to center of top circle of frustum as shown:

$$ \tan \alpha=\frac{\dfrac{R+r}{2}}{(x+h/2)} $$

Perimeter is simply the average at mid-section

$$ p_{mid-section}= \pi (R+r)$$