The polygon area is the region occupied by the polygon. The basic types of polygons are regular polygon and irregular polygon. Learn about the area of a polygon, polygon definition, central point of a polygon, radius of the inscribed circle, circumscribed circle, and polygon of n sides.

## Types of Polygons

Polygon is a closed shape in a two dimensional plane with straight lines. It has an infinite number of sides but all sides are straight lines. The line segments of a polygon are called the sides or edges. The point of intersection of two line segments is called the vertex.

Based on the sides and angles, the polygons are classified into different types. They are regular polygon, irregular polygon, convex polygon, and concave polygon. The regular polygon measures all sides and angles equal. Some of the polygons are triangle, square, pentagon, hexagon, and others.

**Central Point of a Polygon:**

The inscribed and the circumscribed circles of a polygon have the same center, called the central point of the polygon.

**The radius of the Inscribed Circle of a Polygon:**

The length of the perpendicular from the central point of a polygon upon any one of its sides is called the radius of the inscribed circle of the polygon. The radius of the inscribed circle of a polygon is denoted by r.

**The radius of the Circumscribed Circle of a Polygon:**

The line segment joining the central point of a polygon to any vertex is the radius of the circumscribed circle of the polygon. The radius of the circumscribed circle of a polygon is denoted by R.

**Properties of Polygon**

The polygon’s properties are based on the sides and angles. Refer to the below properties and learn entirely about Polygons. They are as follows

- The sum of all interior angles of an n-sided polygon is (n – 2) x 180°.
- The number of triangles formed by joining the diagonals from one corner of a polygon = n – 2.
- The number of diagonals in a polygon with n sides = n(n – 3)/2.
- The measure of each interior angle of n-sided regular polygon = [(n – 2) × 180°]/n
- The measure of each exterior angle of n sided regular polygon = 360°/n

### Area of Regular Polygons Formulas

- If a regular polygon has three sides, then it is called a triangle and its area formula is ½ x base x height
- If the regular polygon has 4 sides, then it called square, and its area formula is side².
- If the regular polygon has 5 sides, then it is called the pentagon and its area formula is /2 x side length × distance from the center of sides to the center of the pentagon.
- If the regular polygon has 6 sides, then it is called the hexagon and its area formula is (3√3)/2 × distance from the center of sides to the center of the hexagon.
- The area of the rectangle formula is length x breadth.
- The area of the rhombus formula is 1/2 x product of diagonals.
- The area of the hexagon is [3(√3)a²/2] square units.
- The area of the octagon is 2a² (1 + √2) square units.
- The area of a polygon is n/2 × a × √(R² – a²/4) square units.

### How to Find the Area of N- Sided Polygons?

Follow the below mentioned simple steps and instructions to calculate the area of a polygon having N-Sides easily.

- If the given is a regular polygon, then substitute the values in the formula.
- If the given figure is an irregular polygon or regular polygon having n sides can use the below process.
- Divide the geometric figure into the combination of regular polygons like triangle, square, rectangle, etc.
- Find the area of each shape.
- Add up those areas to get the polygon area in square units.

### Area of a Polygon Word Problems

**Example 1.**

Find the area of a regular pentagon whose perimeter is 40 units and whose apothem is 5 units?

**Solution:**

Given that,

The perimeter of the regular pentagon = 40 units

Apothem = 5 units

Area of the regular pentagon = ½ (perimeter) (apothem)

= ½ (40) (5)

= 20 x 5 = 100 sq units

Therefore, the area of regular pentagon = 100 sq units.

**Example 2.**

Find the area of a regular octagon each of whose sides measures 25 cm?

**Solution:**

Given that,

Side of a regular octagon a = 25 cm

Area of octagon A = 2a² (1 + √2)

Substitute a = 25 in above equation.

A = 2 x 25² (1 + √2)

= 2 x 625 (1 + √2) = 1250 (1 + √2)

= 1250 x 2.414 = 3,017.76 cm²

Therefore, regular octagon area is 3017.76 cm².

**Example 3.**

The area of a regular pentagon is 215 cm². Find the length of the side of the pentagon?

**Solution:**

Given that,

Regular pentagon area A = 215 cm²

Pentagon area formula is 1/4 ((√(5 (5 + 2 √5) s²))

Pentagon area = 215 cm²

1/4 ((√(5 (5 + 2 √5) s²)) = 215

(√(5 (5 + 2 √5) s²)) = 215 x 4 = 860

s² = 860/(√(5 (5 + 2 √5))

s² = 860/6.88 = 125

s = √125

s = 11.18

Therefore, side of regular pentagon is 11.18 cm.

**Example 4.**

Find the area of a regular polygon having 12 sides and each side length is 5 m?

**Solution:**

Given that,

Number of sides of a polygon n = 12

Side length a= 5 cm

Area of regular polygon A = [n * a² * cot(π/n)]/4

A = [12 x 5² x cot(π/12)]/4

= [12 x 25 x (2 + √3)]/4

= (300 x (2 + √3))/4

= 75 (2 + √3)

= 75 (3.73) = 279.9 cm²

Therefore, the area of a regular polygon is 279.9 cm².