Skill: 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:

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

$$(n-2)180^\circ$$

3. 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.