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, …, P_n-1. Next, let us denote by L_k, k=0,1,2,…,n-1, the distance from P₀ to P_k.

Then, first we see that L₀=0, and that L_k=L_n-k.
Next, let us denote by d_k the ratio
.
Now, consider the triangle with vertices P₀, P₁, and P_k. We name the angle at P_k 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 .

Now, α subtends exactly such an arc (the one from P₀ to P₁). Hence, by the inscribed angle theorem, we find .
Next, we note that β subtends n-k sides of the polygon, and thus an arc of angle . So, we again apply the inscribed angle theorem to get
.
Now, we use the law of sines for these two angles and their opposite sides:
.
Rearranging,
.
so
,
now, since , we see

(this is equivalent to using our earlier property that L_k=L_n-k, and thus d_k=d_n-k).
Note that L₁=L_n-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 d_k 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
,
k=2,3,…,n-2, with d_k=d_n-k.

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Tags: Geometry, Inscribed Angle Theorem, Law of Sines, Math, Monday Math, Regular Polygon, Trigonometry