What is the Meaning of a Polygon?
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.
Properties of a 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
Types of Polygon 3-20
Solved Examples
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.
Solution: Number of sides of a polygon = n (n-3)/ 2 = 54°
n2 – 3n =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.
Quiz Time
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.
Octagon, hexagon, pentagon, and quadrilateral
Pentagon, hexagon, octagon, and quadrilateral
Quadrilateral, pentagon, hexagon, and octagon
Hexagon, pentagon, quadrilateral, and octagon
Interior Angles of a Polygon
The inner angles are created between the neighboring sides within the polygon and are equivalent to each other in the instance of a regular polygon. The number of internal angles is equivalent to the number of sides. The degree of an interior angle of a regular polygon may be computed if the number of sides of the regular polygon is known by applying the given formula:
Interior angle = 180º (n-2) / n, where n is the number of sides
Exterior Angles of a Polygon
Every exterior angle of a regular polygon is created by stretching one side of the polygon either radially or counterclockwise. Then the angle between that expansion and the subsequent side is calculated. Each outside angle of a regular polygon is equivalent and the total of the exterior angles of a polygon equals 360°. An external angle may be computed if the number of sides of a regular polygon is given by using the corresponding formula:
Exterior Angle of a polygon = 360º / n, where n is the number of sides.
Area of Polygons
The area of a polygon is characterized as the amount of space encompassed inside a polygon. The area of polygons may be calculated using several formulae based on if the polygon is a regular or an irregular polygon. For instance, a triangle is a three-sided polygon which is also known as a trigon. The formula for computing the area of the trigon i.e. a triangle is half the product of the base and height of the triangle, i.e. 1/2 * base * height.
Perimeter of Polygons
The perimeter of a polygon is characterized as the distance surrounding a polygon which may be calculated by adding up the length of all provided sides. The perimeter of polygon formula = Sum of all sides. It is stated in sorts of units such as meters, centimeters, feet, etcetera.
Diagonals of a Polygon
Similarly, for an “n” sided-polygon, the count of diagonals may be determined using the following formula:
No. of Diagonals = n(n-3)/2
This equation is basically constructed by the sum of diagonals that every vertex transmits to another vertex and then reducing the total sides. In simpler words, an n-sided polygon contains n-vertices that may be linked with each other in nC2 ways. Now by subtracting n using nC2 methods, the equation derived is n (n-3) / 2.
FAQs on Types of Polygon
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 many 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 for 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) s2
- 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
3. Where are polygons seen in nature?
Polygons arise in rock structures, most typically as the plain sides of crystals. Here in which the angles among the sides. This depends on the type of material from which the crystal is produced. Regular hexagons can also arise when the drying of lava generates sections of closely crowded structures of basalt. This may be observed in the Giant's Causeway in Northern Ireland, or at the Devil's Postpile in California. In biology, the top of the wax beehive formed by bees is an arrangement of hexagons, and the edges and bottom of each cell are likewise polygons.
4. What is the history of polygons?
Polygons have been observed since prehistoric times. The regular polygons were recognized by the ancient Greeks, with the pentagram. Pentagram is a non-convex regular polygon, emerging as back as the 7th century B.C. on a krater by Aristophanes. This was discovered in Caere and housed in the Capitoline Museum. The earliest known detailed investigation of non-convex polygons in particular was done by Thomas Bradwardine in the 14th century. In 1952, Geoffrey Colin Shephard adapted the notion of polygons to the complex plane. Here in each given aspect is joined by an imagined one, to produce complex polygons.
5. What are the components or elements of a polygon?
Side: The side in a polygon is one of the sections presented before that borders the ground of the polygon.
Apex: This is the point wherein two sides of the polygon connect.
Diagonal: This is the section that links two non-adjoining apexes.
Angle: An angle is an opening of two adjoining segments that converge in an apex.
To be able to readily determine the features of a polygon there comes a very basic notation that enables us to understand which component we are talking about at any point. In this manner, we will be able to separate the sides, apexes, or any other part. This can be simply done by naming each corner of the polygon. This would enable each diagonal, apex, or angle to be clearly named and mentioned.