The word polygon takes its origin from the Greek, poly, meaning many and gon, meaning angle. Polygons are 2 dimensional closed shapes made up of straight lines. Sometimes the interior of a polygon is known as its body.
We see polygons all around us. The school building is having a square or rectangular walls. You may come across some houses with triangular roofs. The tiles laid on the floor may be in square or hexagonal shape.
In this chapter, you will learn about polygons and their classifications.
Polygon: It is a closed figure bounded by straight line segments.
Some important facts about polygon:
- Line segments forming a polygon are called sides of the polygon.
- The point where two sides of a polygon meet are called the vertex of the polygon.
- The line segment containing two non-adjacent vertices is called the diagonal of the polygon.
- The angle formed at the vertices inside the closed figure are called interior angles.
Classification of polygons:
Polygons are classified according to the number of sides or vertices they have in to:
- Traingle
- Quadrilateral
- Pentagon
- Hexagon
- Heptagon
- Octagon
- Nonagon
- Decagon
The figure below on the right side shows these polygons.
CONCAVE & CONVEX POLYGONS:
We know that each side of a polygon is connected by two consecutive vertices of the polygon.
A diagonal is a line segment that connects the non-consecutive vertices of a polygon.
If a diagonal lies outside a polygon, then the polygon is called a concave polygon.
If all the diagonals lie inside the polygon, then the polygon is said to be a convex polygon.
A polygon with all its sides equal and all its interior angles equal is said to be a regular polygon. In this chapter, you will learn about regular polygons.
REGULAR & IRREGULAR POLYGONS:
A regular polygon is equiangular and equilateral. The word equiangular means, the interior angles of the polygon are equal to one another. The word equilateral means, the lengths of the sides are equal to one another.
The polygon with unequal sides and unequal angles is called an irregular polygon.
ANGLE SUM PROPERTY:
The sum of all interior angles of a polygon is called the angle sum.
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At one vertex, we extend a side. This side makes an angle with its consecutive side. This angle is called the exterior angle. The interior angle and the exterior angles are adjacent angles. These angles form a linear pair. Hence the sum of the exterior angles of any polygon is 360°.
Let us consider some examples:
Example 1:
Find the sum of all the interior angles of a polygon having 29 sides.
Solution:
We know that sum of all the interior angle in a polygon = (n – 2) × 180°.
Here, n = 29.
Therefore, the sum of all interior angles = (29 – 2) × 180°.
= 27 × 180°.
= 4860°.
Example 2: Is it possible to have a polygon, the sum of whose interior angle is 9 right angles?
Solution:
Let the number of sides be n.
The sum of all interior angles = (2n – 4) × 90°.
So, (2n – 4) × 90° = 9 × 90°.
n = 6.5, hence it is not possible to have a polygon the sum of whose interior angles is 9 right angles.
Example 3: The sides of a Pentagon are produced in order. If the measure of exterior angles so obtained are x°,2 x°,3 x°,4 x°,5 x° and so on, find all exterior angles
Solution:
The sum of exterior angles = 360°.
So, x° + 2x° + 3x° + 4x° + 5x° = 360°.
15x° = 360°.
x° = 24°.
Hence, the exterior angles are 24°, 48°, 72°, 96°, and 120°.
Example 4: If each interior angle of a regular polygon is 144°, Find the number of sides in it.
Solution:
Let the number of sides be n.
Each interior angle = ((2n – 4) × 90°) / n.
144 = 180n – 360°.
n = 10.
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