Interior angles of a polygon formula derivation 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
- Identify the number of sides (n) of the polygon.
Example: Pentagon, n = 5. - Apply the sum formula:
Sum = (n - 2) × 180° = (5 - 2) × 180° = 540°. - 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
FAQs on Interior Angles of a Polygon Explained Clearly
1. What is the sum of the interior angles of a polygon?
The sum of the interior angles of a polygon is given by the formula (n − 2) × 180°, where n is the number of sides.
- This formula works for all simple polygons.
- It is derived by dividing the polygon into triangles.
- Example: For a hexagon (n = 6), the sum = (6 − 2) × 180° = 720°.
2. What is the formula for each interior angle of a regular polygon?
Each interior angle of a regular polygon is calculated using [(n − 2) × 180°] ÷ n.
- This applies only to regular polygons where all angles are equal.
- Example: In a regular pentagon (n = 5), each angle = (3 × 180°) ÷ 5 = 108°.
3. How do you find the number of sides of a polygon from the sum of its interior angles?
You can find the number of sides using the formula n = (S ÷ 180°) + 2, where S is the sum of interior angles.
- Start with (n − 2) × 180° = S.
- Rearrange to get n = (S ÷ 180°) + 2.
- Example: If S = 900°, then n = (900 ÷ 180) + 2 = 5 + 2 = 7 sides.
4. Why is the sum of interior angles (n − 2) × 180?
The formula (n − 2) × 180° works because any polygon can be divided into (n − 2) triangles.
- Each triangle has an angle sum of 180°.
- Dividing a polygon from one vertex creates (n − 2) triangles.
- Multiplying (n − 2) by 180° gives the total interior angle sum.
5. What is the sum of the interior angles of a hexagon?
The sum of the interior angles of a hexagon is 720°.
- Use the formula (n − 2) × 180°.
- For a hexagon, n = 6.
- (6 − 2) × 180° = 4 × 180° = 720°.
6. What is the measure of each interior angle of a regular octagon?
Each interior angle of a regular octagon measures 135°.
- Use the formula [(n − 2) × 180°] ÷ n.
- For n = 8: (6 × 180°) ÷ 8 = 1080° ÷ 8 = 135°.
7. What is the difference between interior and exterior angles of a polygon?
An interior angle is inside the polygon, while an exterior angle is formed outside by extending a side.
- Interior angles add up to (n − 2) × 180°.
- The sum of exterior angles (one at each vertex) is always 360°.
- Each interior and exterior angle at a vertex are supplementary (add up to 180°).
8. Can the sum of interior angles of a polygon be more than 360°?
Yes, the sum of interior angles can be more than 360° for any polygon with more than four sides.
- A quadrilateral has a sum of 360°.
- A pentagon has 540°.
- As the number of sides increases, the angle sum increases using (n − 2) × 180°.
9. How do you calculate the interior angles of a quadrilateral?
The sum of interior angles of a quadrilateral is 360°.
- Use (n − 2) × 180° with n = 4.
- (4 − 2) × 180° = 360°.
- If three angles are known, subtract their sum from 360° to find the fourth angle.
10. What happens to each interior angle as the number of sides increases?
As the number of sides increases in a regular polygon, each interior angle increases and approaches 180°.
- Example: A triangle has 60° (regular).
- A square has 90°.
- A regular 20-sided polygon has 162°.
- As n becomes very large, the polygon resembles a circle and each interior angle gets closer to 180°.
