AREA OF REGULAR POLYGON GIVEN SIDE LENGTH

Find the area of each of regular polygon.

Problem 1 :

Solution :

Number of sides of the given polygon = 6

∠AOB = 360/6 ==> 60

∠COB = 60/2 ==> 30

Apothem = OC

tan θ = BC/OC

tan 30 = 9/OC

0.577 = 9/OC

OC = 9/0.577

OC = 15.59 (Apothem)

Perimeter of the polygon = 618) ==> 108 ft

Area of polygon = (1/2) x Perimeter x apothem

Area of polygon = (1/2) x 108 x 15.59

= 841.86 ft²

Problem 2 :

Solution :

Number of sides of the given polygon = 3

∠AOB = 360/3 ==> 120

∠COB = 120/2 ==> 60

In triangle OCB :

Apothem = OC

tan θ = BC/OC

tan 60 = 4/OC

1.732 = 4/OC

OC = 4/1.732

OC = 2.309

OC = 2.309 (Apothem)

Perimeter of the polygon = 3(4) ==> 12 inches

Area of polygon = (1/2) x Perimeter x apothem

Area of polygon = (1/2) x 12 x 2.309

= 13.854 square inches

Problem 3 :

Solution :

Number of sides of the given polygon = 6

∠AOB = 360/6 ==> 60

∠COB = 60/2 ==> 30

In triangle OCB :

Apothem = OC

tan θ = BC/OC

tan 30 = 6/OC

0.577 = 6/OC

OC = 6/0.577

OC = 10.39

OC = 10.39 (Apothem)

Perimeter of the polygon = 6(12) ==> 72 inches

Area of polygon = (1/2) x Perimeter x apothem

Area of polygon = (1/2) x 72 x 10.39

= 374.04 square meter

Problem 4 :

The game at the right has a hexagon shaped board. Find the area.

Solution :

By observing the hexagon,

the length of apothem = 7.8 inches and side length = 9 inches

Perimeter of hexagon = 6(9)

= 54 inches

Area of hexagon shaped board = (1/2) x perimeter x apothem

= (1/2) x 54 x 7.8

= 27 x 7.8

= 210.6 square inches

So, the area of the hexagon shaped play ground is 210.6 square inches.

Problem 5 :

Each of the tiles in the game is also a regular hexagon. Find the area of one of the tiles if the sides are each 0.9 inches long and each apothem is 0.8 inches long.

Solution :

Side length of hexagon = 0.9 inches

Apothem = 0.8 inches

Number of sides of hexagon = 6

Perimeter of hexagon = 6(0.9)

= 5.4 inches

Area of hexagon shaped board = (1/2) x perimeter x apothem

= (1/2) x 5.4 x 0.8

= 2.7 x 0.8

= 2.16 square inches

So, the area of the hexagon shape is 2.16 square inches.

Problem 6 :

Find the area of the shaped region in the regular polygon at below.

Solution :

Apothem = 5.5 ft and side length = 8 ft

Number of sides = 5

Perimeter = 5(8)

= 40 ft

Area of shaded region = area of pentagon - area of triangle

Base of triangle = 8 ft and height = 5.5 ft

= (1/2) x perimeter x apothem - (1/2) x base x height

= (1/2) x 40 x 5.5 - (1/2) x 8 x 5.5

= 20 x 5.5 - 4 x 5.5

= 110 - 22

= 88 square feet

So, the required area of the shaded region is 88 square feet.

Problem 7 :

Find the area of the shaped region in the regular polygon at the below.

Solution :

Number of sides of polygon = 8

Side length of polygon = 2 meter

Perimeter = 8(2)

= 16 meter

Length of apothem = 2.4 m

Area of polygon = (1/2) x apothem x perimeter

= (1/2) x 2.4 x 16

= 1.2 x 16

= 19.2 square meter

Area of trapezium = (1/2) x height x sum of bases

= (1/2) x 1.4 x (4.8 + 2)

= 0.7 x 6.8

= 4.76 square meter

Area of shaded region = Area of polygon - area of trapezium

= 19.2 - 4.76

= 14.44 square meter

So, the required area of the shaded region is 14.4 square meter.

Problem 8 :

Find the area of the shaped region in the regular polygon at the below.

Solution :

Number of sides = 6

Apothem = 6.9 m and side length of hexagon = 8 m

Perimeter = 6(8)

= 48 m

base of triangle = 8 m and height of triangle = 6.9(2) ==> 13.8 m

Area of shaded region = area of hexagon - area of triangle

Area of hexagon = (1/2) x apothem x perimeter

= (1/2) x 48 x 6.9

= 24 x 6.9

= 165.6 square meter

Area of triangle = (1/2) x base x height

= (1/2) x 8 x 13.8

= 4 x 13.8

= 55.2 square meter

Area of shaded region = 165.6 - 55.2

= 110.4 square meter