Download Trigonometry and Area

9.5 – Trigonometry and Area
Area of a Regular Polygon
Remember from Chapter 7
Theorem 7-12: The area of a regular polygon is
half the product of the apothem and the perimeter.
A = 1/2 a p
a
p
Note that the perimeter (p) is
equal to the length of a side (s)
times the number of sides (n).
p=n•s
Example 1
Find the area of the
regular pentagon.
First find the perimeter.
p=n•s
p = (5) (10)
p = 50 ft
Now find the area.
A=½ap
A = ½ (8) (50)
A = 200 ft2
Using Trigonometry to Find Area
In the previous example, you were given both
the apothem and the length of one side of the
regular pentagon.
What if you only know one of the measurements
and are asked to calculate the area of the polygon?
You can still calculate the area using your
trigonometric functions.
Example 2
Find the area of a regular decagon with the length of
one side equal to 12 cm.
We know the formula to find the area.
A = 1/ 2 a p
And we can easily find the perimeter.
p=n•s
p = 10 • 12
p = 120 cm.
But how can we find the apothem?
Example 2 (continued)
You must first remember from Chapter 7 that the
vertex angle of the isosceles triangle formed by the
radii of a regular polygon has a measure equal to:
360
Vertex  =
n
where n = # of sides
radius
vertex
angle
apothem
Example 2 (continued)
Remember also that the apothem bisects the vertex
angle of the isosceles triangle formed by the radii of a
regular polygon.
360
ACB =
= 36o
10
36
ACM = BCM =
= 18o
2
The apothem also bisects the side of the polygon.
12
AM = MB =
= 6 cm.
2
Example 2 (continued)
Find the area of a regular decagon with
the length of one side equal to 12 cm.
Now find the apothem using the tangent function.
Opposite 6
tan 18 =
=
Adjacent a
o
6
6
a=
=
= 18.47 cm.
o
tan 18
0.3249
And finally, calculate the area.
A=½ap
A = ½ (18.47) (120)
A = 1,108 cm2
Example 3
Find the area of a regular hexagon with a perimeter of
90 ft. Give your answer to the nearest foot.
B
Find the length of one side and segment AD:
p=n•s
90 = 6 • s
15 ft = s
a
15
AD = DC =
= 7.5 cm.
2
Find the measure of the vertex angle?
360
Vertex  = ABC =
= 60o
6
60
ABD = CBD =
= 30o
2
A
D
C
p = 90 ft.
Example 3 (continued)
Find the area of a regular hexagon with a
perimeter of 90 ft. Give your answer to the
nearest foot.
B
a
A
D
C
Next, find the apothem using the tangent function.
Opposite 7.5
tan 30 =
=
Adjacent
a
o
7.5
7.5
a=
=
= 13.0 cm.
o
tan 30
0.5774
Note:
Since this is a 30°-60°-90°
triangle, the apothem (long leg) is
equal to 3 • 7.5 cm (short leg).
Finally, calculate the area.
A=½ap
A = ½ (13.0) (90)
A = 585 ft2
Example 4
Find the area of a regular pentagon
with a radius of 12 m.
B
12
a
Vertex Angle:
A
D
360
Vertex  = ABC =
= 72o
5
72
ABD = CBD =
= 36o
2
Apothem (Long Leg):
a
cos 36 =
12
o
a = 12 cos 36o = 9.7 m
C
Example 4 (continued)
Find the area of a regular pentagon
with a radius of 12 m.
B
12
a
Length of one side of Pentagon:
AD
sin 36 =
12
o
A
D
AD = 12 sin 36o = 7.05 m
AC = AD 2 = 14.1 m
Perimeter:
Area:
p = n • s = 5 • 14.1 = 70.5 m
A = ½ a p = ½ (9.7) (70.5) = 342 m2
C
Theorem 9-1: Area of a Triangle Given SAS
The area of a triangle is one half the product of the
lengths of two sides and the sine of the included angle.
B
c
A
1
Area of ABC = bc (sin A)
2
a
b
C
Example 5
Find the area of a each triangle.
b.
a.
Area
Area
Area
Area
1
= bc (sin A)
2
1
= (36) (60) (sin 27 o )
2
1
= (36) (60) (0.454)
2
= 490 mm 2
1
Area = bc (sin A)
2
1
(9) (18) (sin 60o )
2
1
Area = (9) (18) (0.866)
2
Area = 70 yd 2
Area =
Homework:
p. 500
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