Interior Angles Of A Polygon – Formula And Solved Examples

The concept of interior angles of a polygon plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding this concept helps students quickly determine angles in polygons and solve geometry problems efficiently.

What Is Interior Angle of a Polygon?

An interior angle of a polygon is the angle formed between two adjacent sides of a polygon that meets inside the shape. Polygons such as triangles, quadrilaterals, pentagons, and hexagons all have interior angles. You’ll find this concept applied in areas such as types of angles, geometry problem-solving, and calculating properties of regular and irregular polygons.

Key Formula for Interior Angles of a Polygon

Here’s the standard formula: \( \text{Sum of Interior Angles} = (n - 2) \times 180° \), where ‘n’ is the number of sides of the polygon. For a regular polygon (all sides and angles equal), each interior angle is \( \dfrac{(n - 2) \times 180°}{n} \).

Polygon Types and Angle Table

Polygon	Number of Sides (n)	Sum of Interior Angles	Each Interior Angle (Regular)
Triangle	3	180°	60°
Quadrilateral	4	360°	90°
Pentagon	5	540°	108°
Hexagon	6	720°	120°
Octagon	8	1080°	135°
Decagon	10	1440°	144°

Step-by-Step Illustration: Calculating Interior Angles

1. Identify the number of sides (n) of the polygon. Example: Pentagon, n = 5.
2. Apply the sum formula: Sum = (n - 2) × 180° = (5 - 2) × 180° = 540°.
3. If regular, divide the sum by n for each angle: Each interior angle = 540° ÷ 5 = 108°.

Speed Trick or Vedic Shortcut

To save time in exams, remember: Shortcut: For any polygon with ‘n’ sides, interior angle sum = (n – 2) × 180°. For each angle in a regular polygon, just divide the sum by n. For polygons with large n, practice multiplying quickly, e.g. for n = 12, (12 – 2) × 180 = 1800°.

Real-Life and Exam Application

Whether you’re designing tiles (tessellation), solving geometry questions, or checking building corners, knowing how to calculate the interior angles of a polygon is essential. This topic is tested in boards, JEE Main, and olympiads. Practicing problem-sets with stepwise solutions greatly increases speed and accuracy.

Exterior Angles vs. Interior Angles

Aspect	Interior Angles	Exterior Angles
Definition	Angle inside the polygon	Angle formed by extending one side at a vertex
Sum for n Sides	(n-2) × 180°	360° (always)
Each Angle (Regular)	\((n-2) × 180°/n\)	\(360°/n\)

Stepwise Example Solution

Question: What is the measure of each interior angle in a regular octagon?

1. Number of sides \(n = 8\) 2. Find the sum: \((8 - 2) × 180° = 6 × 180° = 1080°\) 3. Each angle: \(1080° ÷ 8 = 135°\) Answer: Each angle is 135°.

Try These Yourself

• Find the sum of interior angles of a hexagon.
• If each interior angle of a regular polygon is 150°, how many sides does it have?
• What is the measure of each interior angle in a regular dodecagon (12 sides)?
• Show stepwise calculation for the sum of angles in a pentagon.

Frequent Errors and Misunderstandings

• Confusing sum of interior angles (depends on n) and sum of exterior angles (always 360°).
• Forgetting to use brackets: (n-2) × 180°, not n-2 × 180°.
• Mixing regular and irregular polygons when asked for each interior angle.

Relation to Other Concepts

The idea of interior angles of a polygon connects closely with triangle properties (since polygons are divided into triangles for the formula), and with quadrilateral angle sum property. Mastering this helps with more advanced geometry and coordinate geometry topics in higher classes.

Classroom Tip

A quick way to remember: "For any polygon, subtract 2 from the number of sides, and multiply by 180° to get the angle sum." Vedantu’s teachers often show polygons visually—drawing a triangle or quadrilateral, dividing them into triangles—for instant recall of the formula.

We explored interior angles of a polygon—definition, formula, tables, tricks, examples, and common errors. Continue practicing with Vedantu and use the stepwise approach to boost confidence in all geometry topics.

Helpful Study Links

• Types of Polygon
• Angles and Its Types
• Exterior Angles of a Polygon
• Angles in a Pentagon
• Lines and Angles