What is the measure of each interior angle of a regular polygon?

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Answer: To find the measure of an interior angle of a regular polygon, we make use of the formula for each angle = (n - 2) × 180 / n.
Hence, to find the measure of an interior angle of a regular polygon, we use the formula angle = (n - 2) × 180 / n.
Quadrilaterals (Squares, etc)
Because there are 2 triangles in a square ...
The General Rule

A polygon is a two-dimensional geometric figure that has a finite number of sides. The sides of a polygon are made of straight line segments connected to each other end to end. They can have any number of sides. Regular polygons have sides and angles of the same length. Many of the polygons like triangles, squares, and rectangles are used extensively for various purposes.

Answer: To find the measure of an interior angle of a regular polygon, we make use of the formula for each angle = (n - 2) × 180 / n.

The formula calculated is only valid in cases of regular-sided polygons.

Explanation:

We use the standard formula (n - 2) × 180 to find the sum of the interior angles of an n-sided polygon.

To find the measure of one interior angle, we divide the result by n.

For example, let's consider a 3-sided regular polygon, that is, an equilateral triangle. We use the formula, and substitute n with 3.

We get the result to be 180.

Now, if we divide it by n, that is, 3, we get 60, which is the measure of an interior angle of a triangle.

Note: The particular formula is true for only regular polygons, that is, where all the angles are the same, and does NOT cover other cases.

Hence, to find the measure of an interior angle of a regular polygon, we use the formula angle = (n - 2) × 180 / n.

Another example:

Triangles

The Interior Angles of a Triangle add up to 180°

Let's try a triangle:

90° + 60° + 30° = 180°

It works for this triangle

Now tilt a line by 10°:

80° + 70° + 30° = 180°

It still works!
One angle went up by 10°,
and the other went down by 10°

Quadrilaterals (Squares, etc)

(A Quadrilateral has 4 straight sides)

Let's try a square:

90° + 90° + 90° + 90° = 360°

A Square adds up to 360°

Now tilt a line by 10°:

80° + 100° + 90° + 90° = 360°

It still adds up to 360°

The Interior Angles of a Quadrilateral add up to 360°

Because there are 2 triangles in a square ...

The interior angles in a triangle add up to 180° ...

... and for the square they add up to 360° ...

... because the square can be made from two triangles!

A pentagon has 5 sides, and can be made from three triangles, so you know what ...

... its interior angles add up to 3 × 180° = 540°

And when it is regular (all angles the same), then each angle is 540° / 5 = 108°

(Exercise: make sure each triangle here adds up to 180°, and check that the pentagon's interior angles add up to 540°)

The Interior Angles of a Pentagon add up to 540°

The General Rule

Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total:

Shape	Sides	Sum of Interior Angles	Shape	Each Angle
			If it is a Regular Polygon (all sides are equal, all angles are equal)
Triangle	3	180°		60°
Quadrilateral	4	360°		90°
Pentagon	5	540°		108°
Hexagon	6	720°		120°
Heptagon (or Septagon)	7	900°		128.57...°
Octagon	8	1080°		135°
Nonagon	9	1260°		140°
...	...	..	...	...
Any Polygon	n	(n−2) × 180°		(n−2) × 180° / n