HOW TO FIND AREA OF REGULAR POLYGON WITH RADIUS

Radius is the distance between center and one of the vertex.

Area of regular polygon = 1/2 x Perimeter x Apothem

Perimeter = Sum of length of all sides

Apothem is the perpendicular distance between center and one of the sides.

Find the area of each polygon with given radius. Leave the answer in radical.

Problem 1 :

Solution :

Number of sides of the polygon = 4

∠AOB = 360/4 ==> 90

∠COB = 90/2 ==> 45

OB = Radius = 6 cm

In triangle COB,

AB = 2 BC ==> 2(3√2)  ==> 6√2

Area of regular polygon = 1/2 x Perimeter x Apothem

Perimeter = 4 (6√2) ==> 24√2

Area of polygon = (1/2) x 24√2 x 3√2

= 72 cm²

Problem 2 :

Number of sides of the polygon = 3

radius = 18

∠AOB = 360/3 ==> 120

∠COB = 120/2 ==> 60

OB = 18 (radius)

In triangle OCB,

AB = 2(9√3) ==> 18√3

Area of regular polygon = 1/2 x Perimeter x Apothem

Perimeter = 3(18√3) ==> 54√3

Area of regular polygon = 1/2 x 54√3 x 9

= 243√3

Problem 3 :

Solution :

Number of sides of the polygon = 6

radius = 8

∠AOB = 360/6 ==> 60

∠COB = 60/2 ==> 30

OB = 8 (radius)

In triangle OCB,

AB = 2(4) ==> 8

Area of regular polygon = 1/2 x Perimeter x Apothem

Perimeter = 6(8) ==> 48

Area of regular polygon = 1/2 x 48 x 4√3

= 96√3

Problem 4 :

A regular nonagon has a perimeter 45 inches and its apothems are each 6  9/10 inch long.

a) Find the area 

b) Round the length of an apothem to the nearest inch and find the area. How does it compare to the original area ?

Solution :

Perimeter = 45 inches

Length of apothem = 6  9/10 inch ==> 69/10

= 6.9 inches

Approximately 7 inches

Number of sides of nonagon = 9

9(side length of nonagon) = 45

Side length of nonagon = 45/9

= 5

a) Area of nonagon

= (1/2) x perimeter x apothem

= (1/2) x 45 x 6.9

= 155.25 square inches

b) Area of nonagon

= (1/2) x perimeter x apothem

= (1/2) x 45 x 7

= 157.5 square inches

After rounding the length of apothem for the nearest integer, the new area is 2.25

Problem 5 :

The area of a regular octagon is 392.4 square meters, and an apothem is 10.9 meters long

a) Find the perimeter 

b) Find the length of one side.

Solution :

Area of octagon = 392.4 square meter

Length of apothem = 10.9 meter

Number of sides of octagon = 8

(1/2) x perimeter x apothem = 392.4

(1/2) x perimeter x 10.9 = 392.4

Perimeter = 392.4/5.45

= 72

a) So, the perimeter of the octagon is 72 meter.

b) Number of sides x measure of one side = 8(9)

Side length of the octagon = 9 meter

Problem 6 :

Find the approximate area of Fort Jefferson in Dry Tortugas National Park, Florida, if each side is 460 feet long and an apothem is about 398 feet long.

Solution :

Number of sides of the fort = 6

Measure of each side = 460 feet 

Apothem = 398 feet

Area of fort = (1/2) x 6(460) x 398

= 3 x 460 x 398

= 549240 square feet

Problem 7 :

The petals in the flower form a polygon that is approximately a regular pentagon with an apothem 3.4 cm long.

a) Find the approximate area of the pentagon.

b) There are five triangles in the pentagon that are not part of the flower. Assume that they have equal areas and have a height of 1.5 cm. Find the total area of triangles.

c) What is the approximate area of the flower.

Solution :

a) Area of pentagon = (1/2) x perimeter x apothem

perimeter of pentagon = 5(5)

= 25 cm

Apothem = 3.4 cm

Area of pentagon = (1/2) x 25 x 3.4

= 42.5 square meter

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

base = 5 cm and height = 1.5 cm

= (1/2) x 5 x 1.5

= 2.5 x 1.5

= 3.75 square meter

Area of 5 triangles = 5(3.75)

= 18.75 square meter

c) Area of flower = area of pentagon - 5 x area of triangle

= 42.5 - 18.75

= 23.75 square meter