I'm currently going through past exam papers for the history of maths and came across a proof that I can't seem to find online, any help would be greatly appreciated:
Given two regular polygons with $ n $ sides, areas $ p_n $ and $ P_n $, which are inscribed in two circles with diameters $ d $ and $ D $ respectively, then $$ \frac {p_n} {P_n} = \frac {d^2} {D^2} $$ Prove that this equation is true.
Thanks for any help.
On
The polygon could be divided into $n$ triangles from the center point of it. And those triangle has two sides connecting to the center of the polygon to two adjacent vertex of the polygon, with length of $d$.
It is very easy to see that all those triangles from different polygons are homothetic, and thus has area of $kd^2$, where $k$ is a constant for a given $n$. Summing those $n$ triangles' area together, we get the area of polygon, which is $nkd^2$
Thus $$ \frac {p_n} {P_n} = \frac {nk d^2} {nk D^2}=\frac {d^2} {D^2}$$
This not an answer, but too long for a comment so I type it here instead.
First it is good to acknowledge that the question is unnecessarily specific. It says: if I have a [regular polygon with n sides] $A$ and another one, called $B$, which is identical except that the [diameter] of B is exactly [$d/D$] times as long as that of $A$, then the area of $B$ is $[d/D]^2$ times as big that of A.
I put in the square brackets because we could replace the stuff in the brackets with pretty much anything we wanted (given that we get the dimensions right) and the statement would still be true.
The statement: 'if I have a [picture of a dinosaur] $A$ and another one, called $B$, which is identical except that the [funny red strip on the back] of B is exactly $x$ times as long as that of $A$, then the area of $B$ is $x^2$ times as big as that of $A$' is just as correct as the one you are trying to prove.
The general principle is that area scales with the square of length.
Now the reason I find your question very interesting is that this principle is so fundamental that I have no idea how to prove it from even more fundamental principles, unless you somehow go and give a very precise definition of 'area', but e.g. in high school math this never ever happens. People just assume that you have a 'feeling' for what area means and that it has this property.
So in short: I am really curious what other people are going to answer here!