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Quadrilaterals

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CAT 2025 Lesson : Quadrilaterals - Types of Polygons

2.2 Types of Polygons

There are 2 types of polygons

1) Concave Polygons that have at least one interior angle measuring more than

180^{\mathrm{o}}

. In the following figures, the angles marked are the concave angles.

2) Convex Polygons are those that have all angles measuring less than

180^{\mathrm{o}}

Most questions in the exam pertain to convex polygons. Polygons are also classified or referred to basis the number of the sides they have. You should remember the names of

n

-sides figures up to 10 sides.

Number of sides	Name of the polygon	Number of sides	Name of the polygon
4	Quadrilateral	8	Octagon
5	Pentagon	9	Nonagon
6	Hexagon / Sexagon	10	Decagon
7	Heptagon / Septagon

Example 4

In a concave decagon, what is the maximum number of interior angles that could measure more than

180^{\mathrm{o}}

Solution

A decagon is a 10-sided figure.

Sum of interior angles of a decagon

= 180 \times (10 - 2) = 180 \times 8

Here, we can have a maximum of 7 angles that can be more than (or marginally more than)

180^{\mathrm{o}}

, while the last is less than

180^{\mathrm{o}}

.

Answer:

7

2.3 Regular Polygons

In any regular polygon, all its sides are of equal length, all its interior angles are equal and all its exterior angles are equal.

In an

n

-sided regular polygon, each interior angle is the sum of all angles divided by the number of sides.

Interior Angle

= \dfrac{ (n - 2) \times 180^{\mathrm{o}}}{n}

; Exterior Angle

= \dfrac {360^{\mathrm{o}}}{n}

Example 5

In an

n

-sided figure, each angle measures either

145^{\mathrm{o}}

210^{\mathrm{o}}

. If exactly 18 angles measure

145^{\mathrm{o}}

, then

n

= ?

Solution

Number of angles measuring

145^{\mathrm{o}} = 18

(Given in the question)

Number of angles measuring

210^{\mathrm{o}} = n - 18

Sum of angles of the

n

-sided figure

= (n - 2) \times 180^{\mathrm{o}}

⇒

18 \times 145 + ( n - 18 ) \times 210 = (n - 2) \times 180

⇒

2610 + 210n - 3780 = 180n - 360

⇒

30n = 810

⇒

n = 27

Answer:

27

Example 6

If a regular polygon has 27 diagonals, then each of its exterior angles{in degrees) equals?

Solution

Let the regular polygon be an

n

-sided figure.

Number of diagonals

= \dfrac{n (n - 3)}{2} = 27⇒ n(n - 3) = 54

⇒

n (n - 3) = 9(9 - 6)

⇒

n = 9

Each Exterior Angle of the regular nonagon

= \dfrac {360}{9} = 40^{\mathrm{o}}

Answer:

40^{\mathrm{o}}

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