HOW TO FIND THE AREA OF A REGULAR POLYGONS

Find the perimeter area of each regular polygon. Leave your answer in simplest form.

Example 1 :

Name of the shape = pentagon, length of apothem = 7.3 cm and side length = 10.6 cm

Solution :

A pentagon is any polygon that has five sides and five angles.

Number of sides = 5

Perimeter of pentagon = 5 x 10.6

= 53 cm

Perimeter = 53 cm

Length of apothem = 7.3 cm

Example 2 :

Name of the shape = triangle, length of apothem = 14 cm and side length = 28√3 cm

Solution :

Number of sides of triangle = 3

length of apothem = 14 cm and 

side length = 28√3 cm

Perimeter = 3(28√3)

= 84√3 cm

Example 3 :

Name of the shape = 7-gon, length of apothem = 21.8 cm and side length = 21 cm

Solution :

Number of sides = 7

length of apothem = 21.8 cm and 

side length = 21 cm

Perimeter = 7(21)

= 147 cm

Example 4 :

Name of the shape = octagon, length of apothem = 14.1 cm and side length = 11.7 cm

Solution :

Number of sides = 10

length of apothem = 14.1 cm and 

side length = 11.7 cm

Perimeter = 8(11.7)

= 93.6 cm

Example 5 :

You are decorating the top of a table by covering it with small ceramic tiles. The tabletop is a regular octagon with 15-inch sides and a radius of about 19.6 inches. What is the area you are covering?

Solution :

Number of sides of the regular polygon shown is 8.

∠AOB = 360/8

= 45

∠COB = 22.5

tan 22.5 = 7.5/OC

OC = 7.5/tan 22.5

= 7.5/0.414

OC = 18.11(apothem)

Area of polygon = (1/2) x 8(15) x 18.11

= 1086.6

Approximately 1087 square inches.

Area covered by the tiles is 1087 square inches.

Example 6 :

A regular nonagon is inscribed in a circle with a radius of 4 units. Find the area of the nonagon.

Solution :

Number of sides for a nonagon = 9

∠KLM = (360/9)/2

= 20

Side length of the nanogon = 2(1.368)

= 2.736

Perimeter = 9(2.736)

= 24.62

Area of nanogon = (1/2) x 24.62 x 3.758

= 46.26 square units.

So, area of the nanogon is 46.3 square units.

Find the area of the shaded region.

Example 7 :

Solution :

∠AOB = 360/5

= 72

∠COB = 36

BC = 6

Perimeter of pentagon = 5(12)

= 60 inches

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

= (1/2) x 60 x 7.41

= 30 x 7.41

= 222.3 square inches

Area of circle = πr²

= 3.14 x (10.22)²

= 327.96 square inches

Area of shaded region = 327.96 - 222.3

= 105.66

So, the area of the shaded region is 106 square inches.

Find the area of the shaded region.

Example 8 :

Solution :

Area of shaded region = area of circle - area of square

Radius of circle = 14 inches

Side length of square = 14 inches

= πr² - a²

= 3.14(14)² - 14²

=3.14(196) - 196

= 615.44 - 196

= 419.44 square inches

Example 9 :

Describe and correct the error in finding the area of the regular hexagon.

Solution :

Using Pythagorean theorem,

= √15² - 13²

= √225 - 169

= √56

= 7.48

Approximately 7.5 units

Side length of hexagon = 2(7.5)

= 15 units

Perimeter of hexagon = 6(15)

= 90 units

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

= (1/2) x 90 x 13

= 45 x 13

= 585 square units.