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Question

Difficulty: EasyPolygon Angles and Properties

A regular octagon has 8 congruent interior angles. What is the degree measure of one of these interior angles?

Answer: 135 degrees

Answer

The degree measure of one interior angle of a regular octagon is 135 degrees.
The interior angles of a regular octagon sum to 1080∘1080^\circ because (8−2)×180∘=1080∘(8-2) \times 180^\circ = 1080^\circ. Since a regular octagon has 8 equal angles, dividing 1080∘1080^\circ by 88 results in 135∘135^\circ per interior angle.

Step-by-Step Solution

1
Identify the number of sides in a regular octagon.
The number of sides, nn, is 88.
An octagon is defined as a polygon with 8 sides and 8 angles.
2
Calculate the sum of the interior angles of the octagon.
The sum of the interior angles is 1080∘1080^\circ.
The sum of the interior angles of any convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. For n=8n = 8, this is (8−2)×180∘=6×180∘=1080∘(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ.
3
Find the measure of one interior angle by dividing the sum by the number of angles.
Each interior angle measures 135∘135^\circ.
A regular polygon has congruent interior angles. Therefore, dividing the total sum of 1080∘1080^\circ by the 88 congruent angles yields the measure of each individual interior angle: 1080∘÷8=135∘1080^\circ \div 8 = 135^\circ.

Key Concept

The sum of the interior angles of an nn-sided convex polygon is (n−2)×180∘(n-2) \times 180^\circ. For a regular polygon, each interior angle has a measure of (n−2)×180∘n\frac{(n-2) \times 180^\circ}{n}.
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Topics

Polygon Angles and Properties

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