Class 8 - Sum of Angles in a Polygon

The sum of Angles in a polygon is the number of sides a polygon determines the total of its angles. The number of sides in a polygon affects the sum of its angles. There are two main types of angles in every polygon: interior angles (inside the shape) and exterior angles (outside at each corner). We will look into how to find the sum of the angles inside and outside of any polygon.

Table of Contents

• Types of Polygons
• Types of Angles in a Regular Polygon
• Interior Angles Sum of Polygons
• Exterior Angles Sum of Polygons
• Solved Examples on Sum of Angles in a Polygon

What Are Polygon Angles?

Every polygon has angles at each corner where two sides meet. These angles have special properties and patterns.

Types of Polygons

Polygons are classified into various categories depending upon their properties, the number of sides, and the measure of their angles. Based on the number of sides, polygons can be categorized as:

Types of Angles in a Regular Polygon

Regular polygons have special angle properties because all their angles are equal.

Interior Angles in Regular Polygons

Since all interior angles are equal in a regular polygon, we can find the measure of each individual angle.

Formula Sum Of Interior Angle: S = (n - 2) × 180°

Where n = number of sides

Table of Regular Polygon Angles

Exterior Angles in Regular Polygons

An exterior angle is formed when you extend one side of the polygon.

Formula for Each Exterior Angle: Each exterior angle = 360° / n

Where n = number of sides

Each interior angle + its adjacent exterior angle = 180° (they form a straight line)

Relationship Between Interior and Exterior Angles

Formula: Interior angle + Exterior angle = 180°

Example - Regular Hexagon:

Interior angle = 120°

Exterior angle = 60°

Sum = 120° + 60° = 180°

Interior Angles Sum of Polygons

No matter what polygon you have - regular or irregular, convex or concave - the sum of interior angles follows the same formula:

Formula:

Sum of Interior Angles = (n - 2) × 180°

Where n = number of sides

Exterior Angles Sum of Polygons

What is an Exterior Angle?

An exterior angle is formed when you extend one side of the polygon. It's the angle between the extended side and the adjacent side.

Key points:

• Each vertex has one exterior angle
• Exterior angle + its adjacent interior angle = 180°
• They form a linear pair

Why is the Sum Always 360°?

Imagine walking around a polygon Start at one vertex and face along one side. At each corner, you turn through the exterior angle. After going all the way around the polygon, you've made one complete turn.

One complete turn = 360°

So the sum of all the exterior angles = 360°

Irregular Polygons

Even for irregular polygons (where angles are different), the sum of exterior angles is still 360°!

Example - Irregular Quadrilateral:

Exterior angles: 100°, 85°, 95°, 80°

Sum = 100° + 85° + 95° + 80° = 360°

Solved Examples on Sum of Angles in a Polygon

Example 1: Finding Sum of Interior Angles

Question: Find the sum of interior angles of a nonagon (9-sided polygon).

Solution:

• Number of sides (n) = 9

Formula:

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

Calculation:

Sum = (9 - 2) × 180°

Sum = 7 × 180°

Sum = 1260°

Answer: The sum of interior angles in a nonagon is 1260°.

Example 2: Finding Each Interior Angle in Regular Polygon

Question: Find the measure of each interior angle in a regular octagon.

Solution:

Given:

• Regular octagon, so n = 8
• All angles are equal

Step 1: Find total sum

Sum = (8 - 2) × 180° = 1080°

Step 2: Divide by number of angles

Each angle = 1080° / 8 = 135°

Answer: Each interior angle in a regular octagon is 135°.

Example 3: Finding Number of Sides from Angle Sum

Question: The sum of interior angles of a polygon is 2160°. How many sides does the polygon have?

Solution:

• Sum of interior angles = 2160°

Formula:

(n - 2) × 180° = 2160°

Step 1: Divide both sides by 180°

n - 2 = 2160° / 180°

n - 2 = 12

Step 2: Solve for n

n = 12 + 2

n = 14

Answer: The polygon has 14 sides (tetradecagon).

Example 4: Finding Missing Interior Angle

Question: A pentagon has four angles measuring 100°, 110°, 120°, and 95°. Find the fifth angle.

Solution:

Step 1: Find sum of interior angles for pentagon

Sum = (5 - 2) × 180°

Sum = 3 × 180°

Sum = 540°

Step 2: Add the four known angles

100° + 110° + 120° + 95° = 425°

Step 3: Find the missing angle

Missing angle = 540° - 425°

Missing angle = 115°

Answer: The fifth angle is 115°.

Example 5: Finding Each Exterior Angle

Question: Find the measure of each exterior angle of a regular 12-sided polygon (dodecagon).

Solution:

Given:

• Regular dodecagon, n = 12
• All exterior angles are equal

Formula:

Each exterior angle = 360° / n

Each exterior angle = 360° / 12

Each exterior angle = 30°

Sum of all exterior angles = 12 × 30° = 360°

Answer: Each exterior angle is 30°

Conclusion

Understanding angle sums in polygons is like discovering a secret code. Once you know the formulas, you can unlock information about any polygon you encounter. The fact that exterior angles always sum to 360° is one of geometry's most beautiful patterns