*Image source: saigonintela.vn*

What makes polygons so intriguing? It's the incredible world of interior angles! In mathematics, an angle is formed by two rays that share a common endpoint. An interior angle is simply an angle inside a shape. But did you know that the sum of these angles in a polygon is constant, regardless of its size or shape? Let's dive deeper into this fascinating topic.

## Understanding Interior Angles of a Polygon

An interior angle of a polygon is the angle formed by two adjacent sides within the polygon. There are two types of polygons: regular and irregular. A regular polygon has all its interior angles equal in measure, whereas an irregular polygon may have varying interior angle measurements.

To calculate the sum of the interior angles of a polygon, you can use the following formula:

**Sum of Interior Angles of a Polygon = 180 (n-2) degrees**

Here's a table showcasing the number of interior angles and their respective sums for different polygon names:

Polygon Name | Number of Interior Angles | Sum of Interior Angles (degrees) |
---|---|---|

Triangle | 3 | 180° |

Quadrilateral | 4 | 360° |

Pentagon | 5 | 540° |

Hexagon | 6 | 720° |

Septagon | 7 | 900° |

Octagon | 8 | 1080° |

Nonagon | 9 | 1260° |

Decagon | 10 | 1440° |

## Exploring Interior Angles in Different Polygons

### Interior Angles of Triangles

A triangle is a polygon with three sides and three angles. The sum of interior angles in any type of triangle is always 180 degrees. Whether it's an acute, obtuse, or right triangle, the sum of its interior angles remains the same.

### Interior Angles of Quadrilaterals

Quadrilaterals come in various forms, such as squares, rectangles, parallelograms, rhombuses, trapeziums, and kites. All these shapes have four sides and four angles. The sum of interior angles in any quadrilateral is always 360 degrees. For example, a square has interior angles measuring 90 degrees each.

### Interior Angles of a Pentagon

A pentagon consists of five sides and can be formed by joining three triangles side by side. Each triangle has an angle sum of 180 degrees, so the angle sum of three triangles is 540 degrees. Therefore, the angle sum of a pentagon is 540 degrees. In a regular pentagon, each angle measures 108 degrees.

### Interior Angles of Regular Polygons

A regular polygon has all its angles equal in measure. Here are some examples of regular polygons and their corresponding interior angles:

- Triangle: 60°
- Quadrilateral: 90°
- Pentagon: 108°
- Hexagon: 120°
- Septagon: 128.57°
- Octagon: 135°
- Nonagon: 140°
- Decagon: 144°

## Formulas for Interior Angles

There are three formulas to calculate interior angles of polygons:

**Method 1:**
If "n" is the number of sides of a polygon, then the formula is:

**Interior angles of a Regular Polygon = [180°(n) - 360°] / n**

**Method 2:**
If the exterior angle of a polygon is known, the formula to find the interior angle is:

**Interior Angle of a polygon = 180° - Exterior angle of a polygon**

**Method 3:**
If the sum of all the interior angles of a regular polygon is known, the formula to find the interior angle is:

**Interior Angle = Sum of the interior angles of a polygon / n**

Where:

- "n" is the number of sides of the polygon.

## The Polygon Interior Angle Theorem

The Polygon Interior Angle Theorem states that in a polygon with "n" sides, the sum of the interior angles is equal to (2n - 4) × 90°. This theorem can be proven using geometric reasoning and triangle properties.

*Image source: saigonintela.vn*

## Unveiling Exterior Angles

Exterior angles of a polygon are the angles formed at the vertices of the polygon and lie outside the shape. An interesting fact about exterior angles is that the sum of an interior angle and its corresponding exterior angle is always 180 degrees, as they form a linear pair. Additionally, the sum of all the exterior angles of a polygon is always 360 degrees.

The formula to calculate the exterior angle of a polygon is:

**Exterior angle of a polygon = 360 ÷ number of sides**

## Conclusion

The study of interior angles in polygons offers a captivating glimpse into the world of shapes and angles. Whether it's triangles, quadrilaterals, or polygons with numerous sides, the sum of interior angles remains constant. Understanding these angles not only helps in geometry but also enhances our problem-solving skills. So, dive into the world of polygons and explore the fascinating realm of interior angles!

## Related Articles

## Solved Examples

**Q.1: If each interior angle is equal to 144°, then how many sides does a regular polygon have?**

**Solution:**
Given: Each interior angle = 144°

We know that the sum of an interior angle and an exterior angle is 180°.

Exterior angle = 180° - 144° = 36°

The formula to find the number of sides of a regular polygon is:

**Number of Sides of a Regular Polygon = 360° / Magnitude of each exterior angle**

Therefore, the number of sides = 360° / 36° = 10 sides

Hence, the polygon has 10 sides.

**Q.2: What is the value of the interior angle of a regular octagon?**

**Solution:**
A regular octagon has eight sides and eight angles.

Since the sum of interior angles of an octagon is given by (8-2) × 180°, we can calculate:

Sum = (8-2) × 180° = 6 × 180° = 1080°

Since all the interior angles are equal in measure in a regular polygon, the measure of each interior angle is:

Measure of each interior angle = 1080° / 8 = 135°

**Q.3: What is the sum of interior angles in a 10-sided polygon?**

**Solution:**
Given: Number of sides, n = 10

The sum of interior angles can be calculated using the formula:

Sum of interior angles = (10 - 2) × 180° = 8 × 180° = 1440°

## Video Lesson on Angle Sum and Exterior Angle Property

For a more comprehensive understanding of angle sums and the exterior angle property, check out this video lesson:

## Practice Questions

- Find the number of sides of a polygon if each angle is equal to 135 degrees.
- What is the sum of interior angles in a nonagon?

Get ready to explore the marvelous world of interior angles! For further learning, register with BYJU'S - The Learning App and download the app for an easy and enjoyable learning experience.