11.6 –Areas of Regular Polygons

Center of a polygon:

Point equidistant to the vertices of the polygon

center

Radius of a polygon:

Length from the center to the vertex of a polygon

PM

PN

Apothem of the polygon:

Length from the center to the side of a polygon

PQ

Central angle of a regular polygon:

Angle formed by two radii in a polygon

MPN∠

360

n

1. Find the measure of a central angle of a regular polygon with the given number of sides. Round answers to the nearest tenth of a degree if necessary.

6 sides

Central Angle =

360

n=

360

6= 60°

60°

1. Find the measure of a central angle of a regular polygon with the given number of sides. Round answers to the nearest tenth of a degree if necessary.

12 sides

Central Angle =

360

n=

360

12= 30°

1. Find the measure of a central angle of a regular polygon with the given number of sides. Round answers to the nearest tenth of a degree if necessary.

40 sides

Central Angle =

360

n=

360

40= 9°

1. Find the measure of a central angle of a regular polygon with the given number of sides. Round answers to the nearest tenth of a degree if necessary.

21 sides

Central Angle =

360

n=

360

21≈ 17.1°

2. Find the given angle measure for the regular hexagon shown.

Each central angle =

360

n=

360

6= 60°

2. Find the given angle measure for the regular hexagon shown.

m∠EGF = 60°

60°

2. Find the given angle measure for the regular hexagon shown.

m∠EGD = 60°

60°

2. Find the given angle measure for the regular hexagon shown.

m∠EGH = 30°

60°30°

2. Find the given angle measure for the regular hexagon shown.

m∠DGH = 30°

60°30°30°

2. Find the given angle measure for the regular hexagon shown.

m∠GHD = 90°

1

2A san=

Area of a regular polygon:

s = side length

a = apothem length

n = number of sides

3. A regular pentagon has a side length of 8in and an apothem length of 5.5in. Find the area.

1

2A san=

1(8)(5.5)(5)

2A =

(4)(27.5)A =

2110A in=

4. Find the area of the polygon.

Central Angle = _______

Central Angle =

360

n=

360

6= 60°

60°

4. Find the area of the polygon.

c2 = a2 + b2

42 = a2 + 22

16 = a2 + 412 = a2

2 3 a=

2 3

1

2A san=

1(4)(2 3)(6)

2A =

(2)(12 3)A =

224 3A in=

Apothem = __________ 2 3in

5. Find the area of the polygon.

Central Angle = _______

Central Angle =

360

n=

360

5= 72°

72°

5. Find the area of the polygon.

c2 = a2 + b2

6.82 = a2 + 42

46.24 = a2 + 1630.24 = a2

1

2A san=

1(8)(5.5)(5)

2A =

A =110in2

5.5 = a

5.5

Apothem = __________ 5.5in

6. Find the area of the polygon.

m∠ACB = _______ 12cm 12cm

Central Angle =

360

n=

360

3= 120°

120°

60°60°

24cm

30°

6. Find the area of the polygon.

30° 60° 90°

1 3

12

3a =

3 12a =

a 12 b

=3

3

12 3

3=

2

12cm 12cm

4 3

4 3

Apothem = __________ 4 3cm

30°

1

2A san=

1(24)(4 3)(3)

2A =

2144 3A cm=

6. Find the area of the polygon.

4 3

12cm 12cm

30°

m∠ACB = _______

Central Angle =

360

n=

360

6= 60°

60°

30°

5m

7. Find the area of the polygon.

5m60°

7. Find the area of the polygon.

30° 60° 90°

1 3

5 3a =

5 a b

230°

5m5m

5 3

Apothem = __________ 5 3m

60°

1

2A san=

1(10)(5 3)(6)

2A =

A =150 3m2

7. Find the area of the polygon.

30°

5m5m

5 3

8. Find the area of the polygon.

m∠ACB = _______

Central Angle =

360

n=

360

5=72°

72°

36°

8. Find the area of the polygon.

36°

SOH – CAH – TOA

tan 3622

x° =

1

22 tan 36 x° =

= x 15.98

15.9815.98Side Length = __________31.96 cm

8. Find the area of the polygon. Round to two decimal places.

36°

15.9815.98

1

2A san=

A =

12(31.96)(22)(5)

A =1757.8cm2

m∠ACB = _______

Central Angle =

360

n=

360

8=45°

45°

22.5°

9. Find the area of the polygon. Round to two decimal places.

22.5°

SOH – CAH – TOA

cos 22.56

a° =

1

6cos 22.5 a° =

= a 5.54 5.54

9. Find the area of the polygon. Round to two decimal places.

apothem = _______5.54 in

22.5°

SOH – CAH – TOA

sin 22.56

x° =

1

6sin 22.5 x° =

= x 2.3 5.54

2.32.3

9. Find the area of the polygon. Round to two decimal places.

Side length = _______4.6 in

22.5°

5.54

2.3

1

2A san=

1(4.6)(5.54)(8)

2A =

2101.94A in=2.3

9. Find the area of the polygon. Round to two decimal places.