Circle Area
GEOMETRY
NAME_________________________
DATE __________ Per.___________
Area
Find the area of each regular polygon inscribed in the circle of radius r in terms of r, sine, and cosine.
6.
hexagon
7. dodecagon
D is midpoint of BC
mis∠midpoint
D
BAC =of60°
BC
m ∠ BAD = 30°
∆ height AD =rcos30
rt. ∆ leg BD = rsin30
∆ base BC = 2rsin30
area =
m ∠ BAC = 30°
m ∠ BAD = 15°
∆ height AD = rcos15
rt. ∆ leg BD = rsin15
∆ base BC = 2rsin15
area =
A
r
r 2 ⋅ nsin30° ⋅ cos30° B
D
B
r
D
C
A
r 2 ⋅ nsin15°⋅ cos15°
C
Write out the formula in terms of r & n to answer each question:
How do you find the measure of the n-gon’s central angle∠ BAC?
What is formula for the the measure right ∆BDA’s ∠BAD?
Using cos and part (b), express the ∆ height AD in terms of r and n:
Using sin and part (b), express the ∆ base BC in terms of r and n:
What is formula for the area of the regular n-gon (n Isosceles ∆’s)?
(group the r terms first then all the n terms)
8. n-gon
a)
b)
c)
d)
e)
360/n
180/n
r cos( 180 / n )
2 r sin(180/n)
=r2 {n cos(180/n) sin(180/n)}
9.
Fill in the missing boxes in the table below: (Assume r =1)
A
B
C
D
E
F
G
H
1
POLYGON
n=
180/n
∆ height AD
leg BD
∆ base BC
∆ area = bh/2
Area
2
trigon
3
60
0.5000
0.8660
1.7321
0.4330
1.29903811
3
tetragon
4
45
0.7071
0.7071
1.4142
0.5000
2.00000000
4
pentagon
5
36
0.8090
0.5878
1.1756
0.4755
2.37764047
5
hexagon
6
30
0.8660
0.5000
1.0000
0.4330
2.59807488
6
septagon
7
25.7142857
0.9010
0.4339
0.8678
0.3909
2.73640853
7
octagon
8
22.5
0.9239
0.3827
0.7654
0.3536
2.82842525
8
nonagon
9
20
0.9397
0.3420
0.6840
0.3214
2.89254221
9
decagon
10
18
0.9511
0.3090
0.6180
0.2939
2.93892411
10
undecagon
11
16.3636364
0.9595
0.2817
0.5635
0.2703
2.97352226
11
dodecagon
12
15
0.9659
0.2588
0.5176
0.2500
2.99999770
12
fifty-gon
50
3.6
0.9980
0.0628
0.1256
0.0627
3.13332821
13
90-gon
90
2
0.9994
0.0349
0.0698
0.0349
3.13903867
14
100-gon
100
1.8
0.9995
0.0314
0.0628
0.0314
3.13952333
15
180-gon
180
1
0.9998
0.0175
0.0349
0.0174
3.14095205
16
360-gon
360
0.5
1.0000
0.0087
0.0175
0.0087
3.14143051
17
500-gon
500
0.36
1.0000
0.0063
0.0126
0.0063
3.14150732
18
1000-gon
1000
0.18
1.0000
0.0031
0.0063
0.0031
3.14156933
19
million-gon
#####
0.00018
1.0000
3.1416e-6
6.2832e-6
3.14159e-6
3.14159265
π
180
180
r ⋅ nsin  ⋅ cos  = r 2 ⋅ π = πr 2
10. What does n*sin(180/n)*cos(180/n) become as n gets very large?
What is the formula for the area of a circle?
2
n
n