Expert: Paul Klarreich - 2/17/2011

Question
how do i prove that the sum of the interior angles of a convex polygon is 180°(n-2)?

Answer
Questioner: monique
Country: Jamaica
Category: Advanced Math
Private: No
Subject: convex polygon
Question: how do i prove that the sum of the interior angles of a convex polygon is 180°(n-2)?
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I assume you mean:

prove that the sum of the interior angles of a convex polygon WITH n SIDES is 180°(n-2)?

One way is to:

1. name the vertices P[0], P[1],... around the polygon, up to the last one, which I leave to you to name correctly.
2. select one vertex, P[0], and draw diagonals from P[0] to each of the other vertices.

3. determine just how many diagonals you have drawn.
4. count (actually use reasoning) the triangles that result -- show why this number is (n-2).
5. You will use the fact that the sum of the angles of any triangle is 180 degrees.

Over to you now.

BTW, any elementary geometry text will have this proof.