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Determine the sum of interior angles of a polygon

The sum of the interior angles of a simple polygon with nn sides is (n−2)×180∘(n-2)\times180^\circ, because the polygon can be decomposed into n−2n-2 triangles. The total depends only on the number of sides, not the polygon’s size, shape, or regularity, and supports finding unknown angles when the others are known; self-intersecting polygons and more advanced generalized angle treatments are outside this scope.

Detailed Explanation: Determine the sum of interior angles of a polygon

For a simple polygon with nn sides, the sum of its interior angles is

(n−2)×180∘.(n-2)\times 180^\circ.

This works because the polygon can be divided into n−2n-2 triangles. Each triangle has an angle sum of 180∘180^\circ.

Example: Find the sum of the interior angles of an octagon.

  1. An octagon has 88 sides, so n=8n=8.

  2. Substitute 88 into the formula:

(8−2)×180∘ (8-2)\times 180^\circ

  1. Simplify:

6×180∘=1080∘ 6\times 180^\circ=1080^\circ

Therefore, the sum of the interior angles of the octagon is

1080∘.\boxed{1080^\circ}.

The answer depends only on the number of sides, not on the polygon’s size, shape, or whether it is regular.

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    Sum of Inside Angles on a Shape

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