Unit: Understanding Quadrilateral
Chapter: Introduction to Polygon
Reference: – What is a Polygon, Types of Polygons (Regular and Irregular), Convex and Concave Polygons, Classification by Number of Sides, Triangle, Quadrilateral, Pentagon, Hexagon, etc., Interior and Exterior Angles, Sum of Interior Angles of a Polygon, Sum of Exterior Angles of a Polygon, Diagonals of a Polygon, Solved Examples, Odd-One-Out Problems, Common Mistakes
After studying this chapter, you should be able to understand:
- What is a Polygon
- How to Classify Polygons by Number of Sides
- Difference Between Convex and Concave Polygons
- Difference Between Regular and Irregular Polygons
- Calculate Sum of Interior Angles of a Polygon
Introduction to Polygons
Definition
A polygon is a closed two-dimensional figure formed by three or more straight line segments. The line segments are called sides, and the points where two sides meet are called vertices (singular: vertex). The word "polygon" comes from Greek: poly (many) and gon (angles).
When we study polygons, we essentially ask:
"What shape is formed by connecting straight line segments in a closed loop?"
Once we understand polygons, we can classify them, calculate their angles, and explore their properties.
Importance of Polygons
- Found everywhere in nature and design (honeycombs, tiles, buildings)
- Foundation for understanding more complex geometric shapes
- Used in computer graphics, architecture, and engineering
- Helps develop spatial reasoning and problem-solving skills
Example
A triangle is a polygon with 3 sides. A square is a polygon with 4 sides. A pentagon is a polygon with 5 sides. A circle is NOT a polygon because it has curved sides, not straight-line segments.
Subtopics
1. Parts of a Polygon
Sides: The straight-line segments that form the polygon
Vertices: The points where two sides meet (corners)
Diagonals: Line segments joining two non-adjacent vertices
Interior Angles: The angles inside the polygon at each vertex
Exterior Angles: The angles formed between a side and the extension of an adjacent side
2. Convex vs Concave Polygons
Convex Polygon: All interior angles are less than 180°. The polygon bulges outward. Any line segment connecting two points inside the polygon lies entirely inside the polygon.
Example: Square, rectangle, regular pentagon
Concave Polygon: At least one interior angle is greater than 180° (reflex angle). The polygon has a "dent" or "cave" inward.
Example: Arrowhead shape, a star-shaped polygon
Quick Test: If you can draw a line segment between two points inside the polygon that goes outside the polygon, it is concave.
3. Regular vs Irregular Polygons
Regular Polygon: All sides are equal in length, and all interior angles are equal.
Examples: Equilateral triangle, square, regular pentagon, regular hexagon
Irregular Polygon: Sides are not all equal, or angles are not all equal (or both).
Examples: Rectangle (sides not all equal), scalene triangle, any shape that is not regular
4. Sum of Interior Angles of a Polygon
The sum of the interior angles of a polygon depends only on the number of sides (n).
Formula: Sum of interior angles = (n – 2) × 180°
Why this works: A polygon with n sides can be divided into (n – 2) triangles, and each triangle has an angle sum of 180°.
Example 1 – Triangle (n=3): (3 – 2) × 180° = 1 × 180° = 180°
Example 2 – Quadrilateral (n=4): (4 – 2) × 180° = 2 × 180° = 360°
Example 3 – Pentagon (n=5): (5 – 2) × 180° = 3 × 180° = 540°
Example 4 – Hexagon (n=6): (6 – 2) × 180° = 4 × 180° = 720°
5. Measure of Each Interior Angle of a Regular Polygon
Since all interior angles are equal in a regular polygon:
Formula: Each interior angle = [(n – 2) × 180°] / n
Example 1 – Regular pentagon (n=5): (5-2)×180°/5 = 3×180°/5 = 540°/5 = 108°
Example 2 – Regular hexagon (n=6): (6-2)×180°/6 = 4×180°/6 = 720°/6 = 120°
Example 3 – Regular octagon (n=8): (8-2)×180°/8 = 6×180°/8 = 1080°/8 = 135°
6. Sum of Exterior Angles of a Polygon
The sum of the exterior angles of any polygon (convex) is always 360°, regardless of the number of sides.
Important: Take one exterior angle at each vertex (the angle formed by extending one side). The sum is always 360°.
Example: For a pentagon, if you extend each side and measure the exterior angles, they add up to 360°.
7. Each Exterior Angle of a Regular Polygon
Since all exterior angles are equal in a regular polygon:
Formula: Each exterior angle = 360° / n
Relationship: Each interior angle + each exterior angle = 180° (they are supplementary)
Example 1 – Regular pentagon: Each exterior angle = 360°/5 = 72° (check: interior 108° + exterior 72° = 180°)
Example 2 – Regular hexagon: Each exterior angle = 360°/6 = 60° (interior 120° + 60° = 180°)
8. Number of Diagonals in a Polygon
A diagonal connects two non-adjacent vertices.
Formula: Number of diagonals = n(n – 3)/2
Example 1 – Quadrilateral (n=4): 4(4-3)/2 = 4×1/2 = 2 diagonals
Example 2 – Pentagon (n=5): 5(5-3)/2 = 5×2/2 = 5 diagonals
Example 3 – Hexagon (n=6): 6(6-3)/2 = 6×3/2 = 9 diagonals
Solved Examples
Example 1 – Sum of Interior Angles: Find the sum of interior angles of an octagon.
Solution: n = 8, Sum = (8 – 2) × 180° = 6 × 180° = 1080°
Answer: 1080°
Example 2 – Each Interior Angle: Find each interior angle of a regular decagon (10 sides).
Solution: Sum = (10 – 2) × 180° = 8 × 180° = 1440°; Each = 1440°/10 = 144°
Answer: 144°
Example 3 – Each Exterior Angle: Find each exterior angle of a regular hexagon.
Solution: Each exterior angle = 360°/6 = 60°
Answer: 60°
Example 4 – Finding Number of Sides: Each interior angle of a regular polygon is 150°. How many sides does it have?
Solution: Each interior = [(n-2)×180°]/n = 150°
Multiply both sides by n: (n-2)×180 = 150n
180n – 360 = 150n
30n = 360 → n = 12
Answer: 12 sides
Common Mistakes to Avoid
Mistake 1 – Thinking a circle is a polygon
A circle has a curved boundary, not straight line segments.
Correct understanding: Polygons have only straight sides.
Mistake 2 – Using the wrong formula for interior angles
Sum of interior angles = (n – 2) × 180°, not (n – 2) × 360° or n × 180°.
Correct understanding: Memorize the correct formula.
Mistake 3 – Confusing convex and concave
All regular polygons are convex, but not all convex polygons are regular.
Correct understanding: Convex means all interior angles < 180°; concave means at least one angle > 180°.
Mistake 4 – Forgetting that a square is a regular polygon
Some students think only equilateral triangle is regular.
Correct understanding: Any polygon with all sides equal AND all angles equal is regular.
Mistake 5 – Misidentifying a rectangle as regular
A rectangle has all angles 90°, but sides are not all equal (unless it is a square).
Correct understanding: A square is regular; a rectangle that is not a square is irregular.
Mistake 6 – Incorrectly calculating number of diagonals
Using n(n-3) instead of n(n-3)/2 counts each diagonal twice.
Correct understanding: Divide by 2 because each diagonal has two endpoints.
Quick Reference Summary
Polygon: Closed 2D figure with straight sides (3 or more sides)
Convex Polygon: All interior angles < 180° (bulges outward)
Concave Polygon: At least one interior angle > 180° (has a dent)
Regular Polygon: All sides equal AND all angles equal
Sum of Interior Angles: (n – 2) × 180°
Each Interior Angle (Regular): [(n – 2) × 180°] / n
Sum of Exterior Angles: Always 360°
Each Exterior Angle (Regular): 360° / n
Number of Diagonals: n(n – 3)/2
Common Polygons: Triangle (3), Quadrilateral (4), Pentagon (5), Hexagon (6), Octagon (8)