Area of Regular Area of Regular Polygons 5.5Polygons 5.5

• Learn the vocabulary associated with regular polygons.

• Find the area of regular polygons.

Center of a polygonCenter of a polygon – the center of its circumscribed – the center of its circumscribed circle.circle.

Radius of a polygonRadius of a polygon – the radius of its circumscribed – the radius of its circumscribed circle, or the distance from the center to a vertex.circle, or the distance from the center to a vertex.

Apothem of a polygonApothem of a polygon – distance from the center to any – distance from the center to any side of the polygon.side of the polygon.

The area of a regular The area of a regular nn-gon with side length-gon with side length

ss is half the product of the apothem is half the product of the apothem aa and and

the perimeter the perimeter PP..

Find another way to find the area of the regular hexagon shown.

Find the area of one of the triangles and multiply by six.

AFBm.a AFGm.b GAFm.c

In the diagram, ABCDE is a regular pentagon inscribed in circle F. Find each angle measure.

72 36 54

Lets start with an equilateral triangle given an apothem. Find the perimeter and area of the regular polygon.Draw the radius, find the central angle. Divide the central angle in half and solve the triangle.

The central angle is 360 divided by the number of sides. Divide this in half and label it.

6021201203

360

60

Find the side lengths using the 30-60-90 rule.

10

35

Find the perimeter.

35

330P356P Apply the formula.

33052

1A

9.129A375A

Now try an equilateral triangle given a side. Find the perimeter and area of the regular polygon.

Find the side lengths using the 30-60-90 rule.

Find the perimeter.

.cm48P163P Apply the formula.

38162

1A

2cm9.110A364A

Draw the altitude. Solve the triangle.

38sidelong8sideshort

8

38

bh2

1A

Draw the apothem, find the central angle. Divide the central angle in half and solve the triangle.

Find the perimeter.

312P326P Apply the formula.

31222

1A

8.20A312A

Now try an equilateral triangle given a radius. Find the perimeter and area of the regular polygon.

The central angle is 360 divided by the number of sides. Divide this in half and label it.

6021201203

36060

Use the 30-60-90 rule.

32sidelong2sideshort

322

Draw the diagonal, find the area. Use special right triangle to find the side and determine the perimeter.

Find the perimeter.

6.56P240P2104P

20202

1A

200A

Now try a square given a radius. Find the perimeter and area of the regular polygon.

Apply the formula.

Find the side length using the 45-45-90 rule.

210hypotenuse10sidesshort

210

10

Find the perimeter.

56P144P

214A 196A

Now try a square given an apothem. Find the perimeter and area of the regular polygon.

Apply the formula. A = s2

Find the side, find the area, determine the perimeter.

14

Find the perimeter.

42P5.312P Apply the formula.

4235.32

1A

3.127A35.73A

Now try a regular hexagon given a radius. Find the perimeter and area of the regular polygon.

The central angle is 360 divided by the number of sides. Divide this in half and label it.

Use the 30-60-90 rule.

Draw the apothem, find the central angle. Divide the central angle in half and solve the triangle.

30260606

360

3035.3sidelong5.3sideshort 35.3

3.5

Find the perimeter.

328P3

3712P

Apply the formula.

32872

1A

7.169A398A

Now try a regular hexagon given an apothem. Find the perimeter and area of the regular polygon.

The central angle is 360 divided by the number of sides. Divide this in half and label it.

Use the 30-60-90 rule.

Find the central angle. Divide the central angle in half and solve the triangle.

30260606

360

30

3

314hypotenuse

3

37sideshort 3

314

3

37

Find the perimeter.

981.108P8981.1010P Apply the formula.

981.108152

1A 4.817A

Now try a regular pentagon given an apothem. Find the perimeter and area of the regular polygon.

The central angle is 360 divided by the number of sides. Divide this in half and label it.

Use trigonometry.

15

opp36tan

36272725

360

36

Draw the radius, find the central angle. Divide the central angle in half and solve the triangle.

8981.10opposite

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

Find the perimeter P of the table top. An octagon has 8 sides, so P = 8(15) = 120 inches.

So, QS = (QP) = (15) = 7.5 inches.12

12

To find RS, use the Pythagorean Theorem for ∆ RQS.

Find the apothem a. The apothem is height RS of ∆PQR. Because ∆PQR is isosceles, altitude RS bisects QP .

108.18a5.76.196.19a5.7 22222

Apply the formula.

120108.182

1A

A≈ 1086.5 in²

Find the area and perimeter of the regular figure.Find the area and perimeter of the regular figure.

a.a. b.b. c.c.312312 ap 367.32a38.19p 36448 ap

d.d. e.e. f.f. 348324 ap325.2027 ap 3900180 ap

Find the area and perimeter of the regular figure.Find the area and perimeter of the regular figure.

a.a. b.b. c.c.3144a72p 31024a192p 336a36p

d.d. e.e. f.f. 3108a336p 348a324p 312a312p

36 312 332

12 6 18

a.a. b.b. c.c.72a224p 128a232p 288a248p

d.d. e.e. f.f. 1352a2104p 64a32p 800a280p

Find the area and perimeter of the regular figure.Find the area and perimeter of the regular figure.

Find the area and perimeter of the regular figure.Find the area and perimeter of the regular figure.

a.a. b.b. c.c.324a24p 35.13a18p 3384a96p

d.d. e.e. f.f. 4.353a97.69p 35.101a66p 3600a120p

Find the area and perimeter of the regular figure.Find the area and perimeter of the regular figure.

a.a. b.b. c.c.48.153a23.47p 96.1172a47.123p 181a98.48p

AssignmentAssignment

Area & Perimeter of Area & Perimeter of Regular Polygons Regular Polygons