Warm-up

Find the area of the shaded region:

About 66.86 sq cm

20 cm

10

22

L E B

A D M

LAMB is a parallelogram

ADE is an equilateral triangle

About 168.25 sq m

Area of quadrilateral is 784 m²

A.5C

Trig work

8. 13.6

9. 6.4

10. 5.0

11. 19.3

12. 8

13. 7.3

14. 29

15. 52

12 5 12sin ,cos , tan

13 13 5

12 5 5cos ,sin , tan

13 13 12

M M M

L L L

Use a calculator to find each ratio, round to the nearest thousands.

2.tan 87 = 19.081 3. sin 44 =.6947 4. cos 46 =.6947

Use a calculator to find each angle measure to the nearest degree.

5. tan x 0.5774 6. sin x 0.8000 7. cos x 0.7071

30 53 45

1.

16. 59

17. 28.68, 40.96

18. 44 cm

19. 10.46

20. 23.18, 6.21

21. 496.8 ft

22. 264.59 ft

23. 35.33 yds

24. 40.09 feet

1.

What are the formulas for

finding the area of: A = ½ bh

A = bh

A = ½ (b1 +b2)h Rhombus

A = ½ d1 d2

What about other

‘regular’ polygons?

Find the area of three figures –

the hexagon, the pentagon and

the octagon . . .

Think about what you do know . . .

– Use the measurements given, assume the are

regular polygons

11.6: Areas of Regular

Polygons

Class Activity

Area of a regular polygon

11.6 Area of Regular Polygons and

Circles Area of a Regular Polygon: : ______________ where a = _____________ r = ____________ and p = _________________________

*** The apothem is always perpendicular to the side of the polygon!! We are always making ISOSCELES TRIANGLES!!!!

A = ½ ap

apothem radius

perimeter of the polygon

Class Exercises: 1. Find the area of a regular octagon that has a

perimeter of 72 inches.

1. Find the central angle:

.

A B

C

D

E F

G

H

O

P

360/8 = 45

2. Redraw the isosceles triangle: Because the perimeter

is 72 in., we know that each side of our octagon is 9 in.

We also know that OP(the apothem of our polygon) is

the angle bisector, median, and altitude of our triangle

so that…

We do this to find

our apothem, a.

Class Exercises: #1 continued 1. Find the area of a regular octagon that has a

perimeter of 72 inches.

.

A = ½ ap

A = ½ (10.9)(72)

A = 392.4 square inches

3. Solve for a: We will have to use a trig function (SOH

CAH TOA) in order to solve for our apothem…which one?

tan 22.5 = 4.5/a

a = 4.5/tan22.5

a = 10.9

4. Plug in “a” and “p” values into A = ½ap to get the area:

Class Exercises: 2. Find the area of a regular pentagon

whose perimeter is 60 centimeters.

Each side will be 12 cm

The central angles = 72

Tan 36 = 6/a

a = 6/tan 36

a ≈ 8.26

A = ½ ap

A = ½(8.26)(60)

A = 247.7 square cm

36o

12

36o

36o

6

36o

a

The apothem of a regular

hexagon is cm. Find the

perimeter and area of the regular

hexagon.

9 3

Class Exercises: Find the area of the shaded region of m<BAC = m<BCA = 60.

B

C A

O

R

8

Write a Word Equation for what you are trying to do:

Area of the Circle - Area of Triangle = Shaded Area

r = 8

2)8(A

A = 201.06

201.06 -

A = 1/2 ap

Central angle =

360/3 = 120

8/2 = a/1 so a =4

AR = long leg = 4√3

Perimeter = 24√3

A = ½ (4)(24√3)

83.14 = 117.92 units

squared

1 2 √3

In pairs work on Lesson 11.6

Homework

WS 11.6 AREA of Regular Polygons