Polygons

A polygon is a 2D closed figure made of three or more straight line segments. The segments (called sides) meet only at their endpoints (vertices) and never cross.

A polygon looks like a shape formed by joining matchsticks end to end without gaps or overlaps.

Examples:

• Triangle (3 sides)
• Quadrilateral (4 sides)
• Pentagon (5 sides)
• Hexagon (6 sides)
• …and so on.

---

Key Classifications

Based on Sides

• Regular Polygon: All sides and all angles are equal.
• Irregular Polygon: Sides and angles are not all equal.

Based on Convexity

• Convex Polygon: All interior angles < 180°; diagonals lie inside.
• Concave Polygon: At least one interior angle > 180°; at least one diagonal lies outside.

---

Interior Angle and Exterior Angle Properties

Sum of Interior Angles:

$$\text{Sum} = (n - 2) \times 180^\circ$$

Where $n$ = number of sides

Each Interior Angle (in Regular Polygon):

$$\text{Each Angle} = \frac{(n - 2) \times 180^\circ}{n}$$

Each Exterior Angle (in Regular Polygon):

$$\text{Each Angle} = \frac{360^\circ}{n}$$

Exterior Angle Rule: The sum of one exterior angle at each vertex of any polygon = 360°.

---

Diagonals of a Polygon

Number of Diagonals:

$$\text{Diagonals} = \frac{n(n - 3)}{2}$$

• A triangle has 0 diagonals.
• A quadrilateral has 2 diagonals.
• A pentagon has 5 diagonals.

---

Key Formulas Recap

---

Visual Understanding

Example: Regular Pentagon (5 sides)

      A
    /   \
   E     B
   \     /
    D---C

• Interior angle = $\frac{(5-2) \times 180}{5} = 108^\circ$
• Exterior angle = $72^\circ$
• Diagonals = $\frac{5(5-3)}{2} = 5$

Concave Polygon Example:
One vertex pointing inward, creating an internal angle > 180°.

---

Conceptual Tips and Common Mistakes

---

Shortcuts and Tricks

• If asked:
  "Each exterior angle is 30°, how many sides?"
  Use:

  $$n = \frac{360^\circ}{\text{Exterior Angle}} = \frac{360}{30} = 12$$

• Shortcut for quick verification:
  Interior angle + exterior angle = 180°

• To count diagonals quickly:
  Use the table:

  Sides (n)	Diagonals
  3	0
  4	2
  5	5
  6	9
  7	14
  8	20

---

Examples

Example 1:

Find the sum of interior angles of a hexagon.

$$(6 - 2) \times 180 = 720^\circ$$

---

Example 2:

Each exterior angle of a regular polygon is 40°. Find the number of sides.

$$n = \frac{360}{40} = 9$$

---

Example 3:

Find the number of diagonals in a 10-sided polygon.

$$\text{Diagonals} = \frac{10(10 - 3)}{2} = \frac{70}{2} = 35$$

---

Example 4:

What is each interior angle of a regular octagon?

$$\frac{(8 - 2) \times 180}{8} = \frac{1080}{8} = 135^\circ$$