The concept of polygons goes beyond just their sides. Sometimes, you may come across shapes with the same number of sides. So, how do you differentiate them? It's all about the angles!

In this article, we'll explore angles in polygons, including how to find them, the interior and exterior angles, and how to calculate their sizes. By the end, you'll have a clear understanding of this fundamental geometric concept.

## How to Find the Angles of a Polygon?

A **polygon** is a two-dimensional figure with multiple straight-line segments. The sum of the angles inside a polygon is the total measure of all its interior angles. To calculate the angles of a regular polygon, you can use the following formula:

**Sum of interior angles = 180° * (n - 2)**

Here, **n** represents the number of sides of the polygon. Let's look at some examples to see how it works.

### Examples

**Angles of a Triangle:**

A triangle has 3 sides, so n = 3. Substituting this into the formula:

Sum of interior angles = 180° * (n - 2)
= 180° * (3 - 2)
= 180° * 1
= 180°

**Angles of a Quadrilateral:**

A quadrilateral is a 4-sided polygon, so n = 4. By substituting this into the formula:

Sum of interior angles = 180° * (n - 2)
= 180° * (4 - 2)
= 180° * 2
= 360°

Feel free to explore more examples using the same formula for different polygons, such as pentagon, octagon, or even a hectagon.

## Interior Angles of Polygons

The interior angle is the angle formed inside a polygon between two sides. The number of sides in a polygon is equal to the number of angles it has. The formula to calculate the size of each interior angle of a polygon is:

**Measure of each interior angle = 180° * (n - 2)/n**

Again, **n** represents the number of sides. Let's check out a few examples:

**Size of the Interior Angle of a Decagon:**

A decagon has 10 sides, so n = 10. Substituting it into the formula:

Measure of each interior angle = 180° * (n - 2)/n
= 180° * (10 - 2)/10
= 180° * 8/10
= 18° * 8
= 144°

You can also try finding the interior angles of other polygons, such as a hexagon, rectangle, or pentagon, using the same formula.

## Exterior Angles of Polygons

The exterior angle is the angle formed outside a polygon between one side and an extended side. The measure of each exterior angle of a regular polygon is given by:

**Measure of each exterior angle = 360°/n**

Again, **n** represents the number of sides. An important property of regular polygons is that the sum of the measures of their exterior angles is always 360°.

## Examples

**Exterior Angle of a Triangle:**

For a triangle, n = 3. Substituting it into the formula:

Measure of each exterior angle = 360°/n = 360°/3 = 120°

**Exterior Angle of a Pentagon:**

For a pentagon, n = 5. Substituting it into the formula:

Measure of each exterior angle = 360°/n = 360°/5 = 72°

Please note that the interior and exterior angle formulas only work for regular polygons. Irregular polygons have different measures for their interior and exterior angles.

Now, let's explore some example problems to further enhance our understanding of interior and exterior angles of polygons.

**Example 1:**

The interior angles of an irregular 6-sided polygon are 80°, 130°, 102°, 36°, x°, and 146°. Let's calculate the size of angle x in the polygon.

Solution:

For a polygon with 6 sides, n = 6. Substituting it into the formula:

Sum of interior angles = 180° * (n - 2)
= 180° * (6 - 2)
= 180° * 4
= 720°

Therefore, 80° + 130° + 102° + 36° + x° + 146° = 720°

Simplifying the equation: 494° + x = 720°

Subtracting 494° from both sides: 494° - 494° + x = 720° - 494°

x = 226°

**Example 2:**

Find the exterior angle of a regular polygon with 11 sides.

Solution:

n = 11

The measure of each exterior angle = 360°/n = 360°/11 ≈ 32.73°

**Example 3:**

The exterior angles of a polygon are 7x°, 5x°, x°, 4x°, and x°. Determine the value of x.

Solution:

Sum of exterior angles = 360° 7x° + 5x° + x° + 4x° + x° = 360°

Simplifying the equation: 18x = 360°

Dividing both sides by 18: x = 360°/18 x = 20°

Therefore, the value of x is 20°.

**Example 4:**

What is the name of a polygon whose interior angles are each 140°?

Solution:

The size of each interior angle = 180° * (n - 2)/n

Therefore, 140° = 180° * (n - 2)/n

Multiplying both sides by n: 140°n = 180° (n - 2)

140°n = 180°n - 360°

Subtracting both sides by 180°n: 140°n - 180°n = 180°n - 180°n - 360°

-40°n = -360°

Dividing both sides by -40°: n = -360°/-40° = 9

Therefore, the number of sides is 9, making it a nonagon.

## Conclusion

Understanding angles in polygons is essential for geometry. By knowing the formulas to calculate the angles, you can easily determine the measurements of the interior and exterior angles in various polygons. Remember, practice is key when it comes to mastering this concept. Keep exploring examples and solving problems to strengthen your knowledge of angles in polygons.