1. Understanding Interior and Exterior Angles
Whenever two sides of a polygon meet at a vertex, they form an angle. This angle can be viewed from inside the polygon or outside it, giving us two types of angles:
- Interior angle: The angle formed inside the polygon at a vertex.
- Exterior angle: The angle formed outside the polygon when one side is extended.
1.1. Interior Angles
An interior angle is the angle between two adjacent sides, measured inside the polygon. Every vertex of a polygon has one interior angle.
Example: In a rectangle, each interior angle is \(90^\circ\).
1.2. Exterior Angles
An exterior angle is formed when we extend one side of the polygon. The exterior angle and the interior angle at a vertex form a linear pair.
So for every vertex:
\(\text{Exterior angle} = 180^\circ - \text{Interior angle}\)
2. Sum of Interior Angles of a Polygon
The sum of all interior angles of a polygon depends on how many sides the polygon has. We use the idea of splitting the polygon into triangles.
2.1. Deriving the Formula Using Triangles
If a polygon has \(n\) sides, we can divide it into \((n - 2)\) triangles by drawing diagonals from one vertex. Each triangle has an angle sum of \(180^\circ\).
So the total interior angle sum is:
\((n - 2) \times 180^\circ\)
2.2. Examples
- Triangle (\(n=3\)): Sum = \((3-2) \times 180^\circ = 180^\circ\)
- Quadrilateral (\(n=4\)): Sum = \((4-2) \times 180^\circ = 360^\circ\)
- Pentagon (\(n=5\)): Sum = \((5-2) \times 180^\circ = 540^\circ\)
This pattern works for any convex polygon.
3. Sum of Exterior Angles of a Polygon
The exterior angles of a polygon have a very neat property: if we take one exterior angle at each vertex and walk around the polygon, the angle sum is always \(360^\circ\), no matter how many sides the polygon has.
3.1. Walk-Around Explanation
Imagine you walk around a polygon, turning at each vertex to follow the boundary. The total amount you turn to complete the full loop is a complete revolution:
\(360^\circ\)
3.2. Important Result
For any polygon (convex):
\(\text{Sum of exterior angles} = 360^\circ\)
This remains true even for irregular polygons, as long as we take one exterior angle at each vertex.
4. Angles of a Regular Polygon
A regular polygon has all sides equal and all angles equal. Because of this, we can find each interior and exterior angle directly.
4.1. Interior Angle of a Regular Polygon
If a regular polygon has \(n\) sides, then each interior angle is:
\(\displaystyle \dfrac{(n - 2) \times 180^\circ}{n} \)
Example:
- Square (\(n = 4\)): interior angle = \(90^\circ\)
- Regular hexagon (\(n = 6\)): interior angle = \(120^\circ\)
4.2. Exterior Angle of a Regular Polygon
Since the sum of all exterior angles is always \(360^\circ\), and all angles in a regular polygon are equal:
\(\displaystyle \text{Exterior angle} = \dfrac{360^\circ}{n}\)
Examples:
- Equilateral triangle (\(n=3\)): exterior angle = \(120^\circ\)
- Regular hexagon (\(n=6\)): exterior angle = \(60^\circ\)
5. Connecting Interior and Exterior Angles
At every vertex of a polygon:
\(\text{Interior angle} + \text{Exterior angle} = 180^\circ\)
They always form a linear pair. This makes it easy to compute one if the other is known.
5.1. Example
If an interior angle of a polygon is \(135^\circ\), then the exterior angle at that vertex is:
\(180^\circ - 135^\circ = 45^\circ\)