Table Of Contents
What are Polygons?
A polygon is defined as a closed geometric figure formed by finite line segments. The point of intersection of the line segments are called the vertices of the polygon. The line segments of a polygon are also known as the edges or the sides of the polygon. There are two types of polygons:
- Regular Polygon: A polygon with all sides of equal measure.
- Irregular Polygon: A polygon with unequal side lengths.
Angles of Polygons
A polygon has two types of angles, interior and exterior angles.
- Interior angle: An interior angle of a polygon is an angle formed on its inside by any two adjacent sides of the polygon.
Let’s consider an example. Here we have a three sided polygon, a triangle.
∠1, ∠2, and ∠3 are interior angles of the triangle.
- Exterior angle: An exterior angle of a polygon is an angle formed by the side and extension of its adjacent sides. The exterior angles of polygon add up to 360°.
- In the image, ∠4, ∠5, and ∠6 are exterior angles.
- The sum of an interior angle and its corresponding exterior angle is always 180° since they lie on the same straight line.
Solved Examples
Example 1: Find the value of x.
Solution:
The exterior angles are 54°, 77°, 89°, 33°, and x°.
Sum of exterior angles = 360°
54°+77°+ 89°+ 33°+ x=360° [Substitute]
253°+ x=360° [Add]
x=360°-253° [Subtraction property of equality]
x=107° [Simplify]
So, the value of the unknown angle ‘x’ is 107°.
Example 2: What is the value of x?
Solution:
The interior angle is is 53° and the exterior angle and x°.
Sum of the adjacent exterior and the interior angle is 180°.
53°+x=180° [Substitute]
x=180°-53° [Subtraction property of equality]
x=127° [Simplify]
So, the value of the unknown angle ‘x’ is 127°.
Example 3: John draws a polygon with exterior angles
(2x+4)°,(3x-7)°,(4x+3)°,(5x-8)°,(6x-2)°,and (x+13)°. What is the value of x?
Solution:
Exterior angles of the polygon are: (2x+4)°,(3x-7)°,(4x+3)°,(5x-8)°,(6x-2)°,and (x+13)°.
Sum of exterior angles = 360°
On substituting the values we get,
(2x+4)°+(3x-7)°+(4x+3)°+(5x-8)°+(6x-2)°+ (x+13)°=360°
21x+3°=360° [Combine and add like terms]
21x=360°-3° [Subtraction property of equality]
21x=257° [Simplify]
\(x=\frac{257^{o}}{21}\) [Division property of equality]
x=12.23° [Simplify]
So, the value of ‘x’ is 12.23°.
The sum or total of all the exterior angles of a polygon is 360°.
Regular polygons are polygons with equal sides or edges. For example equilateral triangles, squares and many other examples.
The sum of interior angles of an ‘n’ sided polygon is always equal to the product of (n-2) and 180°.
A circle is a curved figure with no straight line as sides. So, a circle is not a polygon.