Geometry

11.4 Areas of Regular Polygons

Definitions

• Regular polygon- a polygon that is equiangular and equilateral.

In the upper right side of your paper, please draw a regular triangle, a regular quadrilateral, a regular hexagon, and a regular octagon.

New words for the vocab list. Also add median of a trapezoid.

Definitions

• Center- the center of the circle that circumscribes the polygon.

Find the center of each polygon, you may or may not want to draw the circumscribed circle.

. center . center . center . center

Definitions

• Radius- the segment from the center to a vertex of the polygon.

Draw one radii of each regular polygon.

. . . . r r

rr

Definitions

• Central angle- the angle formed by two consecutive radii.

Draw one central angle of each regular polygon.

. . . .

Measure of a central angle = 360/n n is the number of sides

360/3

Find the measure of each central angle.

120o

360/4

90o

360/6

60o

360/8

45o

Many opportunities to use your skills of Pythagorean Theorem, 45-45-90, and 30-60-90 right triangles!

Definitions

• Apothem- The distance (perpendicular) from the center to a side

of the polygon.

Draw one apothem of each regular polygon.

. . . . a

aa

a

Area of a Regular Polygon

A = ½ a papothem

perimeter

WHY?

.apothem

x

Area of the green triangle = ½ apothem(x)

x

x

x

x

x

The regular hexagon is made up of 6 green triangles.

Area of the regular hexagon = ½ apothem(6x)

Area of the regular hexagon = ½ apothem(perimeter)

This is true for all regular polygons.

Fill in the table.

24 3

r a p A

1. 8

2.

3. 8

4. 72

1. 2. 3. 4.

.8

45o

90o

. . .4√2

4√2

8√2

P = 4(8√2)

32√2

A = ½ (4√2)(32√2)

128

6√3

6√3

6√3 6√3

3√345o

45o

3√3

3√3

3√6

3√6

A = ½ (3√3)(24√3)A = (3√3)(12√3)

108

A = (2√2)(32√2)

88√2

8√2

816

P = 4(16)

64

A = ½ (8)(64)A = (4)(64)

256

18

18

18 18

9

9

9

9√2

9√2

A = ½ (9)(72)

A = (9)(36)

324

A = ½ a p

Fill in the table.

5. 6. 7. 8.

6 3

9 3

r a p A

5. 86.

7. 88.

.8

. . .360o/3

120o60o

30o4

4

4√38√3

P = 3(8√3)

A = ½ a p

A = ½ (4)(24√3)

24√3

A = (2)(24√3)

48√3

2√3

2√3 2√3

√3

60o

30o

1

1

2

2

A = ½ (1)(6√3)

3√3

60o

30o

816

16

8√316√3

P = 3(16√3)

48√3

A = ½ (8)(48√3)A = (4)(48√3)

192√3

.3√3 3√3

3√3

60o

30o

3/23

3√32

3/23

A = ½ (3/2)(9√3)

27√34

Fill in the table. Please change some of the numbers and cross off the “Side” column.

9. 10. 11.

r a p A

1.

2.

3.5

3

5 2

. . .360o/6

60o

30o5√2

5√225√2

5√62

5√62

A = ½ a p

P = 6(5√2)

30√2

A = ½ (5√6/2)(30√2)A = (5√6/2)(15√2)

75√3

30o√3

1

2

2

2P = 6(2)

12

A = ½ (√3)(12)

6√3

30o5

52

5√32

5√32

5P = 6(5)

30

A = ½ (5√3/2)(30)

A = (5√3/2)(15)

75√32

Word Problems: Who can write these on the board? Find the area of…1) An equilateral triangle with radius 6√3.

2) A regular hexagon with perimeter of 48.

81√3 square units

96√3 square units

Word Problems: Who can write these on the board? Find the area of…3) A square with radius equal to 24.

4) A regular hexagon with apothem equal to 12√3

5) A regular dodecagon(12-sided) with side = r & apothem = s.

1152 square units

864√3 square units

6rs square units

HW

• P 443 (1-22 skip 17)