In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit. Or simply a closed plane figure bounded by three or more line segment to form a closed loop is called a polygon.

Nomenclature:

• The line segments forming the polygon are called sides.
• The point of intersection of two line segments is called a vertex.
• Number of vertices of a polygon is equal to the number of line segments or sides.

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Different types of Polygons:  This is based on the number of sides that the polygon has. Here are few examples:

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Diagonal of a Polygon: A line segment joining any two non-consecutive vertices is called

a diagonal of the polygon.

The dotted lines are diagonals of the shown polygons.

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Interior and Exterior Angles of a Polygon: This is an important concept.

$\angle 1 \ and\ \angle 2$ are interior angles. These are made by the two sides of the polygon.

$\angle 3$ is called an exterior angle. This is formed by extending a side of the polygon as shown in the adjacent figure.

Just by looking at the figure you can tell that

$\angle 2 + \angle 3 = 180^{\circ}$

Hence we can say that:

Exterior Angle + Adjacent Interior Angle $= 180^{\circ}$

Convex Polygon vs Concave Polygon

If the interior angle of the polygon is less than $= 180^{\circ}$, then it is called convex polygon. If you look at any of the polygons shown above, you will see that all the interior angles are less than $= 180^{\circ}$.

But there can be cases where the interior angle of a polygon could be more than $= 180^{\circ}$. Take a look at the adjacent figure. Here you will see that $\angle 1 > 180^{\circ}$ (which is a reflex angle).

Regular Polygon: A polygon that satisfies the following condition is called a regular polygon.

1. All sides are equal
2. All interior angles are equal
3. All exterior angles are equal

For a regular polygon with $latex n &s=1$ sides we have the following:

Each Interior Angle $=$ $[\frac{(2n-4)\times 90^{\circ}}{n}]^{\circ}$

Proof:

A polygon can be divided into $(n-2)$ triangles.

See a few examples in the adjacent figure.

We know that the sum of the angles of a triangle is $= 180^{\circ}$.

Therefore the sum of the interior angles $= (n-2) \times 180 = (2n - 4) \times 90^{\circ}$

$\Rightarrow Interior \ Angle =$ $\frac{(2n-4)\times 90^{\circ}}{n}$

$Each \ Exterior \ Angle =$ $[\frac{360}{n}]^{\circ}$

Proof:

Sum of all Interior Angles + Sum of all Exterior Angles $= n \times 180 = n \times (2 \times 90)^{\circ}$

Sum of all Exterior Angles $= n \times (2 \times 90)^{\circ} - (2n - 4) \times 90^{\circ}=360^{\circ}$

$Each \ Exterior \ Angle =$ $[\frac{360}{n}]^{\circ}$

3. From the above point 2, we can also say that

$n =$ $[\frac{360}{Each \ Exterior \ Angle}]^{\circ}$

Exterior Angle + Adjacent Interior Angle  $= 180^{\circ}$
$\Rightarrow \ Exterior \ Angle=180^{\circ} - Adjacent \ Interior \ Angle$