HOW TO FIND NUMBER OF SIDES OF POLYGON WITH ONE INTERIOR ANGLE

Regular polygon has all sides equal in length and all angles equal in size.

The sum of interior angles of n sided polygon is 

s = (n - 2) x 180°

Measure of each angle= (n-2)×180n

Problem 1 :

Each interior angle of a regular polygon is 174⁰. Find the number of sides of polygon.

Solution :

174= (n-2)×180n174n180 = n - 287n90 = n - 22 = n -87n902 =90n - 87n902 =3n9n3 = 2n = 2(3)n = 6

So, the number of sides of polygon is 6. Then the polygon is known as hexagon.

Problem 2 :

The interior angle of a regular polygon is 135⁰. Work out the number of sides of the polygon.

Solution :

= (n-2)×180n135n180 = n - 227n36 = n - 22 = n -3n42 =4n - 3n42 =n4n4 = 2n = 2(4)n =

So, the number of sides of the regular polygon is 8.

Problem 3 :

The sum of the interior angles in a polygon is 7380⁰. Calculate the number of sides the polygon has.

Solution :

Sum of the interior angles of a polygon = (n - 2) x 180

(n - 2) x 180 = 7380

n - 2 = 7380/180

n - 2 = 41

Add 2 on both sides, we get

n = 41 + 2

n = 43

Problem 4 :

Shown below is a regular pentagon ABCDE

Solution :

Number of sides of given polygon = 5

Sum of interior angles = (n - 2) x 180

= (5 - 2) x 180

= 3 x 180

= 540

Measure of each angle = 540/5

x = 108

In triangle BDC,

DC = BC

∠BDC + ∠DCB + ∠CBD = 180

∠BDC = ∠CBD

y + x + y = 180

2y + x = 180

2y + 108 = 180

2y = 180 - 108

2y = 72

Dividing by 2 on both sides.

y = 72/2

y = 36

Problem 5 :

Shown below is a regular hexagon, with an exterior angle labeled y.

Work out the size of each exterior angle.

Solution :

Number of sides of the polygon above = 6

Sum of the angles of the polygon = (6 - 2) x 180

= 4(180)

= 720

Measure of each angle = 720/6

= 120

Sum of interior angle + y = 180

120 + y = 180

y = 180 - 120

y = 60

Problem 6 :

Shown is a regular hexagon and a regular octagon.

Solution :

Number of sides of hexagon = 6

Number of sides of octagon = 8

Sum of interior angles of hexagon = (6-2)×1806 = 4×1806= 120 Sum of interior angles of octagon = (8-2)×1808 = 6×1808= 135

y = 360 - (120 + 135)

y = 360 - 255

y = 105

Problem 7 :

Shown below is a regular hexagon ABCDEF.

Calculate the angle x.

Solution :

Measure of each angle of hexagon = 120

Since it is regular polygon, CD = BC

∠CDB + ∠CBD + ∠BCD = 180

x + x + 120 = 180

2x + 120 = 180

Subtracting 120 on both sides.

2x = 180 - 120

2x = 60

Dividing by 2, we get

x = 60/2

x = 30

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