Determine exterior angles of regular polygons

An exterior angle of a regular polygon is the angle formed by extending one side, and all such exterior angles are equal because the sides make one complete turn of $360^\circ$ . For an $n$ -sided polygon, each exterior angle measures $360^\circ \div n$ and is supplementary to its corresponding interior angle, connecting the polygon’s side count with its angle measures. The scope is limited to convex regular polygons and degree measures, excluding irregular, star, or more advanced angle cases.

Detailed Explanation: Determine exterior angles of regular polygons

For a convex regular polygon, all exterior angles are equal. As you travel around the polygon, the sides make one complete turn, which is $360^\circ$ .

If a regular polygon has $n$ sides:

\text{Each exterior angle}=\frac{360^\circ}{n}

Example

Find each exterior angle of a regular octagon.

Step 1: Count the sides.

An octagon has $8$ sides, so $n=8$ .

Step 2: Use the formula.

\text{Exterior angle}=\frac{360^\circ}{8}

Step 3: Divide.

\text{Exterior angle}=45^\circ

Therefore, each exterior angle of a regular octagon measures:

\boxed{45^\circ}

The exterior angle and its corresponding interior angle form a straight line, so they are supplementary. Thus, the interior angle would be $180^\circ-45^\circ=135^\circ$ .

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Geometry Polygon Exterior Angles - Regular Polygon and Close-up Image to Angle

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