In Geometry, a polygon is a flat or plane, the two-dimensional closed figure made up of line segments. It does not include any curved side. The word polygon is derived from the Greek language where “poly” means many and ‘gonna’ means ‘angle’. The line segment which is used to make a polygon is known as polygon sides or edges. A minimum of three line segments is needed to draw any closed figure. The corners or points where two line segments meet each other is known as the vertex of a polygon. The classification of a polygon is described on the basis of its number of sides and vertices. For example, a polygon of three sides and three angles is known as a triangle whereas a polygon of four sides and four angles is known as a quadrilateral.

Examples of Polygon

The figures given below are some examples of a polygon

Name the vertices of a polygon.

A vertex is a point where the line segments of a polygon meet each other. It is also known as the corners of a polygon. In a polygon, if there are n sides then there will be n vertices.

For example, Tetrahedron has 4 vertices

The Pentagon has 5 vertices.

What are the different types of polygons?

The different types of the polygon are based on its number of sides and angles. These are:

Regular polygon

Irregular polygon

Convex polygon

Concave polygon

Trigons

Quadrilateral polygon

Pentagon polygon

Hexagon polygon

Equilateral polygon

Equiangular polygon

Let us now discuss the types of polygon individually:

Regular polygon

A regular polygon is a polygon whose all the 6 sides and interior angles are equal. For example, the number of sides of a regular hexagon is 6 and its interior angle is equal to 180 degrees.

Irregular Polygon

A polygon with unequal sides and angles is known as an irregular polygon.

Example

A quadrilateral with unequal sides

An isosceles triangle with two sides equal and the third side with different measurements is also a regular polygon.

Convex Polygon

In convex polygons, the measurement of the interior angles is always less than 180°. The vertex of the convex polygon is always outwards. A convex polygon is exactly opposite to the concave polygon.

Example: An irregular polygon whose vertices are outward.

Concave polygon

In a concave polygon, the measure of any one of the concave polygons is more than 180°. The vertices of the concave polygon are either outwards or inwards.

Trigons

A polygon with 3 sides and 3 vertices is known as trigon. The triangles are further classified into different categories.

Scalene Triangle – A triangle with all the three sides unequal is known as a scalene triangle.

Isosceles triangle- A triangle with any two sides equal is known as an isosceles triangle.

Equilateral triangle- A triangle with all the three equal sides and each angle measured to 60° is known as an equilateral triangle.

Quadrilateral polygon

A polygon with 4 sides and 4 vertices is known as a quadrilateral polygon. The different types of quadrilateral polygons are parallelogram, square, rectangle ,etc.

Pentagon polygon

A polygon with 5 sides and 5 vertices is known as pentagon polygon. The length of all the sides of a pentagon should be equal otherwise it will be considered as an irregular pentagon.

Hexagon polygon

A polygon with six sides and 6 vertices is known as a hexagon polygon. Regular hexagons have 6 sides and all its interior and exterior angles are equal.

Equilateral polygon

A polygon whose sides are equal is known as an equilateral polygon. For example, a square, an equilateral triangle etc.

Equiangular polygon

A polygon whose all interior angles are equal is known as an equiangular polygon. For example, a rectangle is an equiangular polygon.

Here are some of the properties of a polygon

The number of the sides of the polygon shapes

The angles between the sides of the polygon shape

The length of the sides of the polygon shapes.

X sides polygon has x (x-3)/2 diagonals

The sum of all the exterior angles of a polygon is equal to 360 degrees.

The sum of all angles of the x -sided polygon is ( x-20) x 180.

The number of diagonals x- sided polygon has x(x-3)/2 diagonals.

Each angle of a regular polygon is ( x-2) * 180 / x

Here, we will discuss the different names of the polygon 3-20 bases on their sides and the measurement of the angle

1. In triangle ABC, the height is represented by h and its value is 5cm. the base of the triangle is 4 cm. Find the area of the triangle.

Solution: Given,

Base = 4 cm

Height = 5 cm

Area of the triangle = ½ x base x height

Area = ½ x 4 x 5 = 10 cm

Hence, the area of a triangle is 10 cm.

2. In a given quadrilateral ABCD, the side BD is 15 cm and the height of the two triangles ABD and BCD are 5cm and 7 cm respectively. Find the area of quadrilateral ABCD.

Solution: Diagonal of a triangle BD = 15 cm2

Heights, H1 = 5cm and H2 = 7 cm

Sum of the heights of two triangles of a quadrilateral are = 5 + 7 = 12 cm

Hence, the area of a quadrilateral = ½ x diagonal x (Sum of the heights of two triangles)

½ (15 x 12) = 90 cm2

Hence, the area of the quadrilateral is 90 cm2

3. A polygon has 54 diagonals. Find the number of sides in the polygon.

Number of sides of a polygon = n (n-3)/ 2 = 54°

n2 – 3 =108°

n = 12

Hence, the polygon with 54 diagonals has 12 sides.

4. The ratio of the measurement of an interior angle of a regular octagon t the measurement of its exterior angles is:

Solution: Exterior angle = 360/8 = 45

Interior angle = 180 – 45

Interior angle: exterior angle = 135: 45

= 3: 1

Hence, the ratio is 3:1.

1. The sum of the exterior angles of a convex polygon of n sides is equal to

a. 4 right-angle

b. 2/n right angle

c. ( 2n-4) right- angled

d. n/2 right angle.

2 .The sum of all the interior angles of a regular polygon is four times the sum of its exterior angles. Name the type of polygon.

a. Hexagon

b. Triangle

c. Decagon

d. Nonagon

3. Identify the below sets that show the polygon is arranged in the decreasing sequence of the number of sides.

a. Octagon, hexagon, pentagon, and quadrilateral

b. Pentagon, hexagon, octagon, and quadrilateral

c. Quadrilateral, pentagon, hexagon, and octagon

d. Hexagon, pentagon, quadrilateral, and octagon

FAQ (Frequently Asked Questions)

1. What is a simple polygon?

A simple polygon is a closed series of line segments that do not intersect with each other. It means, it has finitely various line segments and each line segment endpoint is shared by two segments and the segments do not intersect each other. In other words, a simple polygon is a polygon whose sides do not intersect with each other. Jordan polygon is another name of a simple polygon. It is also known as Jordan polygon because the Jordan theorem is used to prove that such a polygon divides the plane into two regions i.e. the interior region and the exterior region. A simple polygon in the plane is geometrically equivalent to a circle and its interior part is geometrically equivalent o a disk.

2. Explain the polygon formula.

Polygon is derived from the Greek word where poly means many and gonna means angles. So, we can say that a two-dimensional closed figure with multiple angles is known as a polygon.

There are numerous properties of a polygon based on its sides, diagonals, area, angles, etc. Let us learn how to find these using polygon formulas.

Polygon formula to find area:

Area of a regular polygon

= ½ n sin (360°/ n) s^{2}

Polygon formula to find interior angles

Interior angles of a regular polygon is

(n-2)180°

The formula of a polygon to find triangles

Interiors of triangles in a polygon = (n-2)

Where n represents the number of sides of a triangle and s represents the length from the center to corner