## Monday Math 140

What are the lengths of the diagonals of a regular n-sided polygon with sides of unit length?

Consider any regular n-gon. First, let us label the vertices with P0, P1, …, Pn-1. Next, let us denote by Lk, k=0,1,2,…,n-1, the distance from P0 to Pk.

Then, first we see that L0=0, and that Lk=Lnk.
Next, let us denote by dk the ratio
$d_k=\frac{L_k}{L_1}$.
Now, consider the triangle with vertices P0, P1, and Pk. We name the angle at Pk as α and the angle at P1 as β.

Now, we note that since any regular polygon may be inscribed in a circle, the angles α and β are inscribed angles in the circle. Note that each side of the n-gon is the chord of an arc of angle $\frac{2\pi}n$.

Now, α subtends exactly such an arc (the one from P0 to P1). Hence, by the inscribed angle theorem, we find $\alpha=\frac12\frac{2\pi}n=\frac{\pi}n$.
Next, we note that β subtends nk sides of the polygon, and thus an arc of angle $(n-k)\frac{2\pi}n=2\pi-\frac{2k\pi}n$. So, we again apply the inscribed angle theorem to get
$\beta=\frac12\left(2\pi-\frac{2k\pi}n\right)=\pi-\frac{k\pi}n$.
Now, we use the law of sines for these two angles and their opposite sides:
$\frac{L_1}{\sin\alpha}=\frac{L_k}{\sin\beta}$.
Rearranging,
$\frac{L_k}{L_1}=\frac{\sin\beta}{\sin\alpha}$.
so
$d_k=\frac{L_k}{L_1}=\frac{\sin\left(\pi-\frac{k\pi}n\right)}{\sin\frac{\pi}n}$,
now, since $\sin(\pi-\theta)=\sin\theta$, we see
$d_k=\frac{\sin\frac{k\pi}n}{\sin\frac{\pi}n}$
(this is equivalent to using our earlier property that Lk=Lnk, and thus dk=dnk).
Note that L1=Ln-1 is the length of the side of the n-gon. Thus, for a regular polygon with sides of unit length, L1=1, and then the dk are thus the lengths of the diagonals, and we have our answer; for a regular n-sided polygon with sides of unit length, the diagonals have lengths
$d_k=\frac{\sin\frac{k\pi}n}{\sin\frac{\pi}n}$,
k=2,3,…,n-2, with dk=dnk.

### 3 Responses to “Monday Math 140”

1. Monday Math 141 « Twisted One 151's Weblog Says:

[…] us combine the results from the previous two weeks (here and here). We found that for a regular n-gon with unit sides, the diagonals have lengths , k=2,3,…,n-2, […]

2. jackaljim Says:

Some of the pictures are missing… (can’t be seen)

• twistedone151 Says:

Fixed. I’m in the process of switching posts over to WordPress’ LaTeX from the no-longer working server I previously used.