Exterior Angles of a Polygon
When any side of a polygon is extended, the angle formed between the extended side and the adjacent side is called an exterior angle. Exterior angles are formed outside the polygon.
Every polygon has as many exterior angles as it has sides. At each vertex, the interior angle and the exterior angle together form a linear pair, meaning they add up to 180°.
The most important property of exterior angles is that the sum of all exterior angles of any convex polygon is always 360°, regardless of the number of sides. This result holds for triangles, quadrilaterals, pentagons, hexagons, and all other convex polygons.
What is Exterior Angles of a Polygon?
Definition: An exterior angle of a polygon is the angle formed between one side of the polygon and the extension of an adjacent side.
Key Terms:
- Exterior Angle: The angle formed outside the polygon when a side is extended.
- Interior Angle: The angle formed inside the polygon between two adjacent sides.
- Linear Pair: Interior angle + Exterior angle = 180° at each vertex.
- Convex Polygon: A polygon where all interior angles are less than 180°. The exterior angle sum property applies to convex polygons.
- Regular Polygon: A polygon with all sides equal and all angles equal. Each exterior angle of a regular polygon with n sides = 360°/n.
Important: At each vertex, only ONE exterior angle is considered (not both possible extensions). We take one exterior angle at each vertex.
Exterior Angles of a Polygon Formula
Formula 1: Exterior Angle Sum Property
Sum of all exterior angles of any convex polygon = 360°
Formula 2: Each exterior angle of a regular polygon
Each exterior angle = 360° / n
Where:
- n = number of sides of the regular polygon
Formula 3: Relationship between interior and exterior angles
Interior angle + Exterior angle = 180°
Formula 4: Number of sides from exterior angle
n = 360° / (each exterior angle)
Derivation and Proof
Proof that the sum of exterior angles of any convex polygon is 360°:
Method 1: Using interior angle sum
- The sum of interior angles of a polygon with n sides = (n - 2) x 180°.
- At each vertex, Interior angle + Exterior angle = 180°.
- There are n vertices, so the sum of all (Interior + Exterior) pairs = n x 180°.
- Sum of all interior angles + Sum of all exterior angles = n x 180°.
- (n - 2) x 180° + Sum of exterior angles = n x 180°.
- Sum of exterior angles = n x 180° - (n - 2) x 180°.
- Sum of exterior angles = 180° x [n - (n - 2)].
- Sum of exterior angles = 180° x 2 = 360°.
Method 2: Walking around the polygon
- Imagine walking along the sides of a convex polygon.
- At each vertex, you turn by the exterior angle.
- After completing one full trip around the polygon, you return to the starting point facing the same direction.
- This means you have made one complete revolution = 360°.
- The total turning = sum of all exterior angles = 360°.
Types and Properties
Exterior angles appear in different types of polygons:
1. Equilateral Triangle (n = 3):
- Each interior angle = 60°
- Each exterior angle = 180° - 60° = 120°
- Sum = 3 x 120° = 360°
2. Square (n = 4):
- Each interior angle = 90°
- Each exterior angle = 180° - 90° = 90°
- Sum = 4 x 90° = 360°
3. Regular Pentagon (n = 5):
- Each exterior angle = 360°/5 = 72°
- Each interior angle = 180° - 72° = 108°
4. Regular Hexagon (n = 6):
- Each exterior angle = 360°/6 = 60°
- Each interior angle = 180° - 60° = 120°
5. Regular Octagon (n = 8):
- Each exterior angle = 360°/8 = 45°
- Each interior angle = 180° - 45° = 135°
6. Regular Decagon (n = 10):
- Each exterior angle = 360°/10 = 36°
- Each interior angle = 180° - 36° = 144°
Solved Examples
Example 1: Example 1: Exterior angle of a regular hexagon
Problem: Find each exterior angle of a regular hexagon.
Solution:
Given:
- Regular hexagon, so n = 6
Using the formula:
- Each exterior angle = 360° / n
- = 360° / 6
- = 60°
Answer: Each exterior angle of a regular hexagon is 60°.
Example 2: Example 2: Finding number of sides
Problem: Each exterior angle of a regular polygon is 40°. Find the number of sides.
Solution:
Given:
- Each exterior angle = 40°
Using the formula:
- n = 360° / (each exterior angle)
- n = 360° / 40°
- n = 9
Answer: The polygon has 9 sides (it is a nonagon).
Example 3: Example 3: Sum of exterior angles of a pentagon
Problem: Four exterior angles of a pentagon are 70°, 80°, 60°, and 90°. Find the fifth exterior angle.
Solution:
Given:
- Four exterior angles: 70°, 80°, 60°, 90°
- Sum of all exterior angles = 360°
Calculation:
- Sum of four angles = 70° + 80° + 60° + 90° = 300°
- Fifth exterior angle = 360° - 300°
- = 60°
Answer: The fifth exterior angle is 60°.
Example 4: Example 4: Interior angle from exterior angle
Problem: Each exterior angle of a regular polygon is 30°. Find the interior angle and number of sides.
Solution:
Given:
- Each exterior angle = 30°
Step 1: Find interior angle
- Interior angle = 180° - 30° = 150°
Step 2: Find number of sides
- n = 360° / 30° = 12
Answer: The polygon has 12 sides (dodecagon) with each interior angle = 150°.
Example 5: Example 5: Can the exterior angle be 50°?
Problem: Can 50° be an exterior angle of a regular polygon?
Solution:
- If each exterior angle = 50°, then n = 360° / 50° = 7.2
- Since n must be a whole number, 50° cannot be an exterior angle of a regular polygon.
Answer: No, 50° cannot be an exterior angle of a regular polygon because 360 is not exactly divisible by 50.
Example 6: Example 6: Exterior angles of an irregular polygon
Problem: The exterior angles of a quadrilateral are x°, 2x°, 3x°, and 4x°. Find x and all the angles.
Solution:
Given:
- Exterior angles: x°, 2x°, 3x°, 4x°
Using exterior angle sum property:
- x + 2x + 3x + 4x = 360°
- 10x = 360°
- x = 36°
The exterior angles are:
- x = 36°
- 2x = 72°
- 3x = 108°
- 4x = 144°
Verification: 36° + 72° + 108° + 144° = 360°
Answer: x = 36°. The exterior angles are 36°, 72°, 108°, and 144°.
Example 7: Example 7: Finding exterior angle of an equilateral triangle
Problem: Find the measure of each exterior angle of an equilateral triangle.
Solution:
Given:
- Equilateral triangle: all interior angles = 60°
Method 1: Using linear pair
- Exterior angle = 180° - 60° = 120°
Method 2: Using formula
- Each exterior angle = 360° / 3 = 120°
Answer: Each exterior angle = 120°.
Example 8: Example 8: Maximum number of right exterior angles
Problem: What is the maximum number of exterior angles that can be right angles (90°) in a convex polygon?
Solution:
- Sum of all exterior angles = 360°
- If each right exterior angle = 90°, then maximum number = 360° / 90° = 4
- If there are 4 exterior angles of 90°, sum = 360° (which is exactly the total)
- So the polygon must be a quadrilateral with all exterior angles = 90° (a rectangle/square)
Answer: The maximum number of right exterior angles is 4 (in a rectangle).
Example 9: Example 9: Comparing exterior angles of two polygons
Problem: The exterior angle of a regular polygon A is twice the exterior angle of a regular polygon B. If polygon B has 12 sides, find the number of sides of polygon A.
Solution:
Given:
- Polygon B: n = 12
Step 1: Find exterior angle of B
- Exterior angle of B = 360° / 12 = 30°
Step 2: Find exterior angle of A
- Exterior angle of A = 2 x 30° = 60°
Step 3: Find sides of A
- n = 360° / 60° = 6
Answer: Polygon A has 6 sides (a regular hexagon).
Example 10: Example 10: Finding unknown angle using exterior angle sum
Problem: The exterior angles of a hexagon are 50°, 65°, 55°, 70°, 75°, and x°. Find x.
Solution:
Given:
- Five exterior angles: 50°, 65°, 55°, 70°, 75°
Using the exterior angle sum property:
- 50 + 65 + 55 + 70 + 75 + x = 360
- 315 + x = 360
- x = 360 - 315
- x = 45°
Answer: The unknown exterior angle is 45°.
Real-World Applications
Architecture and Construction: Architects use exterior angle properties to design buildings with polygonal floor plans. Calculating the turning angle at each corner requires knowledge of exterior angles.
Navigation and Surveying: When a surveyor walks around a plot of land and measures the turning angle at each corner, the sum of these turning angles (exterior angles) must be 360° for a closed polygon. This serves as a verification check.
Computer Graphics: Drawing polygons on screen requires calculating the exterior angle to determine the rotation at each vertex. Regular polygon drawing algorithms use the formula 360°/n directly.
Road Design: Roundabouts and road junctions designed in polygonal shapes use exterior angle calculations to ensure smooth traffic flow around corners.
Logo and Pattern Design: Many logos and patterns use regular polygons. Designers use exterior angle calculations to space vertices evenly around a circle.
Robotics: Programming a robot to walk along a polygonal path requires specifying the turning angle at each corner, which is the exterior angle.
Key Points to Remember
- An exterior angle is formed when a side of a polygon is extended.
- At each vertex: Interior angle + Exterior angle = 180° (linear pair).
- The sum of all exterior angles of any convex polygon is always 360°.
- This property is true for ALL convex polygons, regardless of the number of sides.
- Each exterior angle of a regular n-sided polygon = 360° / n.
- To find the number of sides: n = 360° / (each exterior angle).
- The exterior angle must be a factor of 360 for a regular polygon to exist.
- As the number of sides increases, each exterior angle decreases.
- A regular polygon cannot have an exterior angle greater than 120° (minimum 3 sides: 360°/3 = 120°).
- The smallest possible exterior angle approaches 0° as n approaches infinity (the polygon approaches a circle).
Practice Problems
- Find each exterior angle of a regular polygon with 15 sides.
- The exterior angles of a quadrilateral are (x+5)°, (2x+10)°, (3x-15)°, and (x+20)°. Find x and all exterior angles.
- Can 25° be an exterior angle of a regular polygon? Give reason.
- Each interior angle of a regular polygon is 156°. Find the number of sides.
- Three exterior angles of a pentagon are 65°, 80°, and 45°. The remaining two angles are equal. Find each.
- How many sides does a regular polygon have if each exterior angle is one-fifth of a right angle?
- The ratio of exterior angle to interior angle of a regular polygon is 1:5. Find the number of sides.
- Find the sum of interior angles of a regular polygon whose each exterior angle measures 24°.
Frequently Asked Questions
Q1. What is the sum of exterior angles of a polygon?
The sum of all exterior angles of any convex polygon is always 360°. This is true regardless of the number of sides.
Q2. How do you find the exterior angle of a regular polygon?
Each exterior angle of a regular polygon = 360° divided by the number of sides (n). For example, a regular pentagon has each exterior angle = 360°/5 = 72°.
Q3. What is the relationship between interior and exterior angles?
At each vertex of a polygon, the interior angle and exterior angle form a linear pair. They add up to 180°. So, Exterior angle = 180° - Interior angle.
Q4. Why is the sum of exterior angles always 360°?
If you walk along the boundary of a polygon, at each vertex you turn by the exterior angle. After one complete trip, you return to the start facing the same direction. This is one full turn = 360°.
Q5. Can an exterior angle of a polygon be greater than 180°?
In a convex polygon, every exterior angle is less than 180°. In a concave polygon, a reflex interior angle can produce an exterior angle that is negative (if measured conventionally), but for NCERT Class 8, we deal with convex polygons only.
Q6. How do you find the number of sides from the exterior angle?
Divide 360° by the measure of each exterior angle: n = 360° / (exterior angle). The result must be a whole number for the polygon to exist as a regular polygon.
Q7. What is the exterior angle of a square?
Each interior angle of a square is 90°, so each exterior angle = 180° - 90° = 90°. Alternatively, 360°/4 = 90°.
Q8. Can 15° be an exterior angle of a regular polygon?
Yes. n = 360°/15° = 24. Since 24 is a whole number, a regular polygon with 24 sides has each exterior angle = 15°.
Q9. What is the minimum and maximum possible exterior angle?
The minimum number of sides is 3 (triangle), giving maximum exterior angle = 360°/3 = 120°. As the number of sides increases, the exterior angle decreases and approaches 0° but never reaches it.
Q10. Do exterior angles apply to concave polygons?
The exterior angle sum of 360° applies to convex polygons. For concave polygons, some exterior angles are considered negative when the vertex is reflex, but the algebraic sum still equals 360°.
