Interior Angles of a Polygon
A polygon is a closed figure made up of straight line segments. Triangles, quadrilaterals, pentagons, and hexagons are all polygons with 3, 4, 5, and 6 sides respectively.
Every polygon has interior angles — the angles formed inside the polygon at each vertex. The sum of these interior angles depends on the number of sides of the polygon.
In Class 7, you learned that the sum of interior angles of a triangle is 180°. In this chapter, you will extend this result to find the sum of interior angles of any polygon using a simple formula.
The key idea is that any polygon can be divided into triangles by drawing diagonals from one vertex. Since each triangle contributes 180° to the total angle sum, the formula becomes straightforward.
What is Interior Angles of a Polygon?
Definition: An interior angle of a polygon is the angle formed between two adjacent sides of the polygon, measured inside the polygon.
Key terms:
- Polygon — A closed plane figure with three or more straight sides.
- Regular polygon — A polygon in which all sides are equal and all interior angles are equal.
- Irregular polygon — A polygon in which sides and angles are not all equal.
- Convex polygon — A polygon where all interior angles are less than 180°.
- Concave polygon — A polygon where at least one interior angle is greater than 180°.
Interior angle sum property: The sum of all interior angles of a polygon with n sides is given by the formula (n − 2) × 180°.
This formula works because any polygon with n sides can be divided into (n − 2) triangles, and each triangle has an angle sum of 180°.
Interior Angles of a Polygon Formula
Formula 1: Sum of Interior Angles
Sum of interior angles = (n − 2) × 180°
Where:
- n = number of sides of the polygon
Formula 2: Each Interior Angle of a Regular Polygon
Each interior angle = (n − 2) × 180° / n
This formula applies ONLY to regular polygons (where all angles are equal).
Quick Reference Table:
- Triangle (n = 3): Sum = 180°, Each angle = 60°
- Quadrilateral (n = 4): Sum = 360°, Each angle = 90°
- Pentagon (n = 5): Sum = 540°, Each angle = 108°
- Hexagon (n = 6): Sum = 720°, Each angle = 120°
- Octagon (n = 8): Sum = 1080°, Each angle = 135°
- Decagon (n = 10): Sum = 1440°, Each angle = 144°
Derivation and Proof
Derivation of the Interior Angle Sum Formula:
Step 1: Take any polygon with n sides.
Step 2: Choose any one vertex of the polygon.
Step 3: Draw diagonals from this vertex to all non-adjacent vertices.
Step 4: Count the number of triangles formed.
From one vertex, you can draw diagonals to (n − 3) other vertices (you cannot draw a diagonal to itself or to the two adjacent vertices). This creates (n − 2) triangles.
Step 5: Each triangle has an angle sum of 180°.
Step 6: The sum of angles of all triangles = Sum of interior angles of the polygon.
Therefore:
- Sum of interior angles = (n − 2) × 180°
Verification for known polygons:
- Triangle (n = 3): (3 − 2) × 180° = 1 × 180° = 180° ✓
- Quadrilateral (n = 4): (4 − 2) × 180° = 2 × 180° = 360° ✓
- Pentagon (n = 5): (5 − 2) × 180° = 3 × 180° = 540° ✓
Types and Properties
Interior angle problems can be classified into the following types:
1. Finding the sum of interior angles:
- Given the number of sides, apply (n − 2) × 180°.
- Example: Sum of interior angles of a heptagon (7 sides) = (7 − 2) × 180° = 900°.
2. Finding each interior angle of a regular polygon:
- Use (n − 2) × 180° / n.
- Example: Each angle of a regular hexagon = (6 − 2) × 180° / 6 = 120°.
3. Finding the number of sides from the angle sum:
- Given the sum S, use n = (S / 180) + 2.
- Example: If sum = 1260°, then n = (1260 / 180) + 2 = 7 + 2 = 9 sides.
4. Finding a missing angle:
- Find the total sum using the formula, then subtract the known angles.
5. Finding the number of sides from each interior angle:
- If each interior angle of a regular polygon is given, solve (n − 2) × 180° / n = given angle.
Solved Examples
Example 1: Example 1: Sum of interior angles of a pentagon
Problem: Find the sum of interior angles of a pentagon.
Solution:
Given:
- Number of sides (n) = 5
Using the formula:
- Sum = (n − 2) × 180°
- Sum = (5 − 2) × 180°
- Sum = 3 × 180°
- Sum = 540°
Answer: The sum of interior angles of a pentagon is 540°.
Example 2: Example 2: Each interior angle of a regular octagon
Problem: Find each interior angle of a regular octagon.
Solution:
Given:
- n = 8 (octagon)
- It is a regular polygon (all angles equal).
Step 1: Sum of interior angles = (8 − 2) × 180° = 6 × 180° = 1080°
Step 2: Each interior angle = 1080° / 8 = 135°
Answer: Each interior angle of a regular octagon is 135°.
Example 3: Example 3: Finding a missing angle in a quadrilateral
Problem: Three angles of a quadrilateral are 75°, 90°, and 120°. Find the fourth angle.
Solution:
Given:
- Three angles: 75°, 90°, 120°
- n = 4 (quadrilateral)
Step 1: Sum of interior angles = (4 − 2) × 180° = 360°
Step 2: Fourth angle = 360° − (75° + 90° + 120°)
- = 360° − 285°
- = 75°
Answer: The fourth angle is 75°.
Example 4: Example 4: Finding number of sides from angle sum
Problem: The sum of interior angles of a polygon is 1440°. How many sides does it have?
Solution:
Given:
- Sum of interior angles = 1440°
Using the formula:
- (n − 2) × 180° = 1440°
- n − 2 = 1440° / 180°
- n − 2 = 8
- n = 10
Answer: The polygon has 10 sides (it is a decagon).
Example 5: Example 5: Finding number of sides from each interior angle
Problem: Each interior angle of a regular polygon is 156°. Find the number of sides.
Solution:
Given:
- Each interior angle = 156°
Using the formula:
- (n − 2) × 180° / n = 156°
- (n − 2) × 180° = 156n
- 180n − 360 = 156n
- 180n − 156n = 360
- 24n = 360
- n = 15
Answer: The polygon has 15 sides.
Example 6: Example 6: Sum of interior angles of a nonagon
Problem: Find the sum of interior angles of a nonagon (9 sides).
Solution:
Given:
- n = 9
Using the formula:
- Sum = (9 − 2) × 180°
- Sum = 7 × 180°
- Sum = 1260°
Answer: The sum of interior angles of a nonagon is 1260°.
Example 7: Example 7: Missing angle in a pentagon
Problem: Four angles of a pentagon are 100°, 115°, 105°, and 110°. Find the fifth angle.
Solution:
Given:
- Four angles: 100°, 115°, 105°, 110°
Step 1: Sum of interior angles of pentagon = (5 − 2) × 180° = 540°
Step 2: Sum of known angles = 100° + 115° + 105° + 110° = 430°
Step 3: Fifth angle = 540° − 430° = 110°
Answer: The fifth angle is 110°.
Example 8: Example 8: Each interior angle of a regular polygon (12 sides)
Problem: Find each interior angle of a regular dodecagon (12 sides).
Solution:
Given:
- n = 12
Step 1: Sum = (12 − 2) × 180° = 10 × 180° = 1800°
Step 2: Each angle = 1800° / 12 = 150°
Answer: Each interior angle of a regular dodecagon is 150°.
Example 9: Example 9: Can a regular polygon have each angle 170°?
Problem: Is it possible to have a regular polygon with each interior angle equal to 170°?
Solution:
Using the formula:
- (n − 2) × 180° / n = 170°
- (n − 2) × 180 = 170n
- 180n − 360 = 170n
- 10n = 360
- n = 36
Since n = 36 is a whole number, yes, it is possible. It would be a 36-sided regular polygon.
Answer: Yes, a regular polygon with each interior angle 170° has 36 sides.
Example 10: Example 10: Interior angle vs exterior angle relationship
Problem: Each interior angle of a regular polygon is 5 times its exterior angle. Find the number of sides.
Solution:
Given:
- Interior angle = 5 × Exterior angle
- Interior angle + Exterior angle = 180° (linear pair)
Step 1: Let exterior angle = x. Then interior angle = 5x.
- 5x + x = 180°
- 6x = 180°
- x = 30°
Step 2: Each exterior angle = 30°.
- Sum of exterior angles = 360°
- Number of sides = 360° / 30° = 12
Answer: The polygon has 12 sides.
Real-World Applications
Architecture and Construction:
- Architects use interior angles when designing polygonal rooms, domes, and floor patterns. Hexagonal tiles fit together perfectly because each interior angle of a regular hexagon is 120° and three hexagons meet at a point (3 × 120° = 360°).
Tiling and Tessellations:
- Only regular polygons with interior angles that divide 360° exactly can tile a flat surface — equilateral triangles (60°), squares (90°), and regular hexagons (120°).
Engineering:
- Bolt heads and nuts are often hexagonal. The 120° interior angle provides a good grip and even force distribution.
Nature:
- Honeycomb cells are hexagonal — the 120° interior angle allows bees to use the least wax for the most storage space.
Computer Graphics:
- Game designers use polygon angle calculations to render 3D shapes, create maps, and design environments.
Road Signs:
- Stop signs are regular octagons (each angle = 135°). Their distinctive shape is recognisable even from behind because of the unique angle pattern.
Key Points to Remember
- The sum of interior angles of a polygon with n sides = (n − 2) × 180°.
- Each interior angle of a regular polygon = (n − 2) × 180° / n.
- A polygon with n sides can be divided into (n − 2) triangles from one vertex.
- The formula works for all convex polygons.
- Interior angle + Exterior angle = 180° at each vertex (they form a linear pair).
- Sum of exterior angles of any convex polygon = 360°.
- As the number of sides increases, each interior angle of a regular polygon approaches 180° but never reaches it.
- A regular polygon cannot have an interior angle of 180° or more.
- To find number of sides from angle sum: n = (Sum / 180) + 2.
- Only three regular polygons can tessellate a plane: triangle (60°), square (90°), and hexagon (120°).
Practice Problems
- Find the sum of interior angles of a heptagon (7 sides).
- Each interior angle of a regular polygon is 140°. Find the number of sides.
- Three angles of a quadrilateral are 80°, 100°, and 95°. Find the fourth angle.
- Find each interior angle of a regular polygon with 20 sides.
- The sum of interior angles of a polygon is 2340°. How many sides does it have?
- Can a regular polygon have each interior angle equal to 100°? Justify.
- The interior angles of a hexagon are x°, (x + 10)°, (x + 20)°, (x + 30)°, (x + 40)°, and (x + 50)°. Find x.
- Each interior angle of a regular polygon is 4 times the exterior angle. Find the number of sides.
Frequently Asked Questions
Q1. What is the formula for the sum of interior angles of a polygon?
The sum of interior angles of a polygon with n sides is (n − 2) × 180°. For example, a pentagon (5 sides) has angle sum = (5 − 2) × 180° = 540°.
Q2. Why do we subtract 2 in the formula (n − 2) × 180°?
Because a polygon with n sides can be divided into (n − 2) triangles by drawing diagonals from one vertex. Each triangle has an angle sum of 180°, so the total is (n − 2) × 180°.
Q3. What is the difference between interior and exterior angles?
An interior angle is formed inside the polygon between two adjacent sides. An exterior angle is formed between one side and the extension of the adjacent side. At each vertex, interior angle + exterior angle = 180°.
Q4. How do you find each interior angle of a regular polygon?
Divide the sum of interior angles by the number of sides: Each angle = (n − 2) × 180° / n. For a regular hexagon, each angle = (6 − 2) × 180° / 6 = 120°.
Q5. Can the interior angle of a regular polygon be 180°?
No. If each interior angle were 180°, the sides would form a straight line and the figure would not be a closed polygon. The maximum value of each interior angle approaches 180° as n increases but never reaches it.
Q6. How do you find the number of sides if the angle sum is given?
Use n = (Sum / 180) + 2. For example, if the sum is 900°, then n = (900 / 180) + 2 = 5 + 2 = 7 sides.
Q7. Does the interior angle formula work for concave polygons?
The formula (n − 2) × 180° gives the sum of interior angles for convex polygons. For concave polygons, reflex angles (greater than 180°) need special treatment, but the formula still holds if interior angles are measured correctly.
Q8. What is the sum of interior angles of a triangle and a quadrilateral?
Triangle: (3 − 2) × 180° = 180°. Quadrilateral: (4 − 2) × 180° = 360°. These are the two most commonly used results.
Q9. Why can only some regular polygons tile a flat surface?
For tiling, the interior angle must divide 360° exactly, so that whole copies fit around a point without gaps. Only 60° (triangle), 90° (square), and 120° (hexagon) satisfy this condition.
Q10. What is the relationship between interior and exterior angles of a polygon?
At each vertex, interior angle + exterior angle = 180°. The sum of all exterior angles of any convex polygon is always 360°, regardless of the number of sides.
