Determine interior angles of regular polygons

A regular polygon has congruent interior angles, so the measure of each angle is found by dividing the interior-angle sum, $(n-2)180^\circ$ , by the number of sides $n$ , equivalently $180^\circ-\frac{360^\circ}{n}$ . The reasoning distinguishes the sum of all interior angles from the measure of one angle and connects polygon structure with algebraic formulas; it applies to common convex regular polygons, not irregular, concave, or more advanced generalized cases.

Detailed Explanation: Determine interior angles of regular polygons

A regular polygon has all sides and all interior angles equal. To find the measure of one interior angle:

Let $n$ be the number of sides.
Find the sum of all interior angles:

(n-2)180^\circ

Divide that sum by $n$ , because all $n$ angles are congruent:

\text{Each interior angle}=\frac{(n-2)180^\circ}{n}

Example: Find each interior angle of a regular octagon.

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

Step 1: Find the sum of the interior angles.

(8-2)180^\circ=6(180^\circ)=1080^\circ

Thus, all eight interior angles together measure $(1080^\circ)$ .

Step 2: Divide by the number of angles.

\frac{1080^\circ}{8}=135^\circ

Therefore, each interior angle of a regular octagon measures

\boxed{135^\circ}

You can also use the equivalent formula:

180^\circ-\frac{360^\circ}{n}

For the octagon:

180^\circ-\frac{360^\circ}{8} =180^\circ-45^\circ =135^\circ

Remember: $((n-2)180^\circ)$ gives the sum of all interior angles, so you must divide by $n$ to find one angle.

Learn by doing: Determine interior angles of regular polygons

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

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