Inner angles in polygonsA point of a polygon is called convex if it's inner angle is less than 180 degrees. Prove that in every simple polygon there is at least one convex point.

Step 1The sum of the inner angles of a simple polygon with n sides is ( n − 2 ) π.Step 2If all inner angles were greater than or equal to π, then that sum would be at least n π, which it isn't. Hence, at least one inner angle is less than π.This argument actually implies that there are at least two convex vertices.

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High school geometry
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2022-08-18

Inner angles in polygons
A point of a polygon is called convex if it's inner angle is less than 180 degrees. Prove that in every simple polygon there is at least one convex point.

Answer & Explanation

Brogan Navarro

Beginner2022-08-19Added 24 answers

Step 1
The sum of the inner angles of a simple polygon with n sides is

(n - 2) π

.
Step 2
If all inner angles were greater than or equal to

π

, then that sum would be at least

n π

, which it isn't. Hence, at least one inner angle is less than

π

.
This argument actually implies that there are at least two convex vertices.

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