Q.1 Each interior angle of a polygon is 108º. How many sides does it have?

Solution: We can solve this problem in two ways

Method: 01

We know, the sum of an interior and an exterior angle in a polygon is always 180º

Given: Each interior angle of a polygon is 108º

∴ 108º+exterior angle =180º

Exterior angle=180º-108º

Exterior angle=72º

So, the number of side in the polygon is \frac{360º}{72º}=5

Method: 02 Using interior angle formula

The measure of each interior angle of a regular polygon of n sides is given \frac{(n-2)×180º}{n}

where n is the number of side of the polygon

Given: Each interior angle of a polygon is 108º

Let the number of sides in the polygon is n

∴ 108º=\frac{(n-2)×180º}{n}

108n=(n-2)×180º

108n=180n-360

-72n=-360

n=5

Thus, the number of side in the polygon is 5

Q.2 Find the sum of interior angles of a polygon having 15 sides.

Solution:

We know, The measure of each interior angle of a regular polygon of n sides is given \frac{(n-2)×180º}{n}

Substitute n=15

Each interior angle = \frac{(15-2)×180}{15}

Each interior angle = \frac{13×180}{15}

Each interior angle= 156º

so, the sum of all interior angles of the polygon = 15×156=2340º

Q.3 In a regular polygon, each interior angle is thrice of the exterior angle. Find the number of sides of the polygon.

Solution:

Let the exterior angle of the regular polygon is x

then, each interior angle will be 3x

We know, the sum of an interior and an exterior angle in a polygon is always 180º

∴ x+3x=180

4x= 180

x= 45º

Since the exterior angle of the polygon, the measure is 45º

Number of side in the polygon is \frac{360}{45}=8

Thus, the required polygon is a regular octagon

Q.4 What is the exterior angle with 11 sided polygon?

Solution:

Each exterior angle of a regular polygon is given by \frac{360}{n}

substitute n=11

Exterior angle= \frac{360}{11}

Exterior angle= 32.72º

So, measure of each exterior angle is 32.72º

Q.5 How many sides do a regular polygon have whose interior angle is 4 times its exterior angle?

Solution:

Let the exterior angle of the regular polygon is x

then, each interior angle will be 4x

We know, the sum of an interior and an exterior angle in a polygon is always 180º

∴ x+4x=180

5x= 180

x= 36º

Since the exterior angle of the polygon, the measure is 36º

The number of sides in the polygon is \frac{360}{36º}=10

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