A polygon is a closed shape made from straight line segments. Polygons are classified by the number of sides they have and by the properties of their angles.

 

 

Regular and Irregular Polygons

A regular polygon has:
• all sides equal
• all interior angles equal

 

Examples include an equilateral triangle, a square and a regular hexagon.

 

An irregular polygon does not have all sides and angles equal.

 

Most polygons encountered in real life are irregular.

 

Regular polygons have symmetry, irregular polygons may not

 

 

Interior and Exterior Angles

At each vertex of a polygon, there is:
• an interior angle inside the shape
• an exterior angle formed by extending one side of the polygon

 

The interior and exterior angle at a vertex form a straight line.

 

 

Sum of an Interior and Exterior Angle

At any vertex of a polygon, the interior angle and its corresponding exterior angle add up to
\( 180^\circ \)

 

This is because they lie on a straight line.

 

This rule applies to all polygons, regular or irregular.

 

 

Sum of Exterior Angles of a Polygon

The sum of the exterior angles of any polygon, taking one exterior angle at each vertex and turning in the same direction, is always
\( 360^\circ \)

 

This is true for:
• triangles
• quadrilaterals
• all other polygons

 

It does not depend on the number of sides or whether the polygon is regular or irregular.

 

The total turning around the shape is one full turn

 

 

Sum of Interior Angles of a Polygon

The sum of the interior angles of a polygon depends on the number of sides.

 

For a polygon with n sides, the sum of the interior angles is given by:

$$
(n - 2) \times 180^\circ
$$

 

Examples:

 

A triangle has:
$$
(3 - 2) \times 180^\circ = 180^\circ
$$

 

A quadrilateral has:
$$
(4 - 2) \times 180^\circ = 360^\circ
$$

 

A pentagon has:
$$
(5 - 2) \times 180^\circ = 540^\circ
$$

 

This formula applies to both regular and irregular polygons.

 

 

Key Points to Remember

Polygons are closed shapes made from straight lines.
Regular polygons have equal sides and equal angles.
Interior and exterior angles at a vertex add up to \( 180^\circ \).
The sum of exterior angles of any polygon is \( 360^\circ \).
The sum of interior angles of a polygon is \( (n - 2) \times 180^\circ \).

 

Understanding polygon angle properties allows you to find unknown angles and analyse shapes confidently.