How to Find Area of Regular Polygon With Side Length

Problem 1 :

What is the area of regular nanagon with 10 cm sides.

Solution :

Number of sides of Nanagon = 9

∠PCR = 360/9 ==> 40

Drawing perpendicular bisector CS, we get

∠PCS = 40/2 ==> 20 and PS = PR/2 ==> 5 cm

The area of the regular nonagon is about 618 cm².

Problem 2 :

Octagon with side length 6 cm

Solution :

Number of sides = 8

∠AOB = 360/8 ==> 45

∠COB = 45/2 ==> 22.5

Side length = 6 cm

Length of an apothem,

Perimeter = sides × length

= 8 × 6

P = 48 cm

Area = 1/2 × aP

= 1/2 × 7.246 × 48

A = 173.91 cm²

The area of the regular octagon is about 174 cm².

Problem 3 :

Decagon with side length 4 yd

Solution :

Number of sides = 10

Side length = 4 yd

Length of an apothem,

AC = 2, BC = 2, Apothem = OC

∠AOB = 360/10 ==> 36

∠COB = 36/2 ==> 18

OC = Apothem

tan θ = BC/OC

tan 18 = 2/a

a = 2 / tan 18

a = 2/0.325

a = 6.153

Perimeter = sides × length

= 10 × 4

P = 40 yd

Area = 1/2 × aP

= 1/2 × 6.153 × 40

A = 123.06 yd²

The area of the regular decagon is about 123 yd².

Problem 4 :

The Pentagon in Arlington, Virginia, is one of the world’s largest office buildings. It is a regular pentagon, and the length of each of its sides is 921 ft. what is the area of land that the Pentagon covers to the nearest thousand square feet?

Solution :

Number of sides = 5

Side length = 921 ft

∠AOB = 360/5 ==> 72

∠COB = 72/2 ==> 36

Length of an apothem,

tan θ = BC/OC

tan 36 = 921/a

a = 921/1.452

a = 634.3

Perimeter = sides × length

= 5 × 921

P = 4605 ft

Area = 1/2 × aP

= 1/2 × 634.3 × 4605

A = 1460475.75 ft²

The area of the regular pentagon is about 1460475 ft².