About Definition of Polygon
- What are Polygons?
- Types of Polygons
- Classify Polygons Based on the Sides Measure and Angles
- Sum of the Interior Angles of a Polygon
- Sum of the Exterior Angles of a Polygon
- Measure of each Interior Angle of a Regular Polygon
- Measure of each Exterior Angle of a Regular Polygon
- Solved Examples
- Frequently Asked Questions
What are Polygons?
Polygons are closed 2D figures made up of line segments. Polygons are flat or plane shapes with no curves. The line segments of the polygon are known as sides or edges as well. The points where the sides of a polygon meet are called the vertices. The following image has shapes that are examples of polygons.
Types of Polygons
Based on the number of sides of a polygon, there are different types of polygons with unique names. Here is a list:
A polygon can have a minimum of three sides as that is the minimum number to create a closed figure.
Note: The smallest polygon is a triangle.
Classify Polygons Based on the Sides Measure and Angles
Based on the measure of the sides, polygons can be classified as regular and irregular polygons.
Regular polygons have equal sides and angles. For example, a square and an equilateral triangle.
Irregular polygons have unequal sides and angles. The following can be treated as examples.
Based on angles, polygons can be classified as convex and concave polygons.
Convex polygons are polygons with all interior angles less than 180°. For convex polygons, all the diagonals are inside a figure. Here are a few examples.
In a concave polygon, there is at least one interior angle greater than 180°. All of the diagonals are not in the interior of the figure. Here are a few examples.
Sum of the Interior Angles of a Polygon
If we consider a polygon with ‘n’ sides, then the formula for the sum of the interior angles of a polygon will be,
(n – 2) × 180°.
Sum of the Exterior Angles of a Polygon
For any polygon with ‘n’ sides, the sum of the exterior angles of polygons is 360°. This is relevant for all kinds of polygons, irrespective of the number of sides they have.
Measure of each Interior Angle of a Regular Polygon
If we consider a regular polygon with ‘n’ sides, then the measure of the interior angles will be, \(\left[\frac{(n-2)180^\circ}{n}\right]\).
Measure of each Exterior Angle of a Regular Polygon
For a regular ‘n’ sided polygon, the measure of exterior angles will be \(\frac{360^\circ}{n}\).
Solved Examples
Example 1: Find the sum of the interior angles of a decagon.
Solution:
As we have learned, a decagon has 10 sides. The formula to find the sum of interior angles is (n – 2) × 180°.
Interior angle sum = (n – 2) x 180°
⇒ (10 – 2) x 180° Substitute 10 for the value of ‘n’
⇒ 8 x 180°
⇒ 1440°
So, the sum of the interior angles of a decagon is 1440°.
Example 2: David has drawn a few figures for his class assignments. Can you help him identify the polygons from his drawings?
Solution:
As we know, a polygon has a closed shape, made up of line segments that join at vertices and is a flat shaped object.
Here, shape 1 is not a closed shape, shape 3 has a curved surface, shape 4 is a 3D figure, and shape 5 is not a closed figure. Shape 2 and shape 6 are polygons, as they are closed shapes with line segments, and are 2D figures.
Example 3: The sum of the interior angles of a polygon is 1620°. How many sides does the polygon have?
Solution:
As we have learned, the sum of the interior angles of a polygon = (n – 2) x 180°
Here, we have to find the value of ‘n’. Hence, substituting the values in the formula,
1620° = (n – 2) x 180° Write the formula
9 = (n – 2) Divide each side by 180°
11 = n Add 2 each side
So, the polygon has 11 sides.
A triangle is the smallest polygon.
When a polygon does not cross over itself and has a single boundary, it is called a simple polygon. Otherwise, it is known as a complex polygon.
Polygons are closed figures made of line segments. A circle has a curved side. Hence, a circle is not a polygon.
An equilateral triangle and square.