Transcript 10-5 Trigonometry and Area

Trigonometry and Area
LESSON 10-5
Additional Examples
Find the area of a regular polygon with 10 sides and side
length 12 cm.
Find the perimeter p and apothem a, and then find
the area using the formula A = 1 ap.
2
Because the polygon has 10 sides and each side
is 12 cm long, p = 10 • 12 = 120 cm.
Use trigonometry to find a.
Because the polygon has 10 sides, m
ACB = 360 = 36.
10
CA and CB are radii, so CA = CB. Therefore, ACM
BCM by the HL
1
1
Theorem, so m ACM = 2 m ACB = 18 and AM = 2 AB = 6.
HELP
GEOMETRY
Trigonometry and Area
LESSON 10-5
Additional Examples
(continued)
tan 18° =
6
a
Use the tangent ratio.
6
a = tan 18°
Solve for a.
Now substitute into the area formula.
A = 1 ap
2
6 .
1
A = 2 • tan 18° • 120
360
A = tan 18°
360
18
The area is about 1108 cm2.
HELP
Substitute for a and p.
Simplify.
Use a calculator.
Quick Check
GEOMETRY
Trigonometry and Area
LESSON 10-5
Additional Examples
The radius of a garden in the shape of a regular pentagon is
18 feet. Find the area of the garden.
Find the perimeter p and apothem a, and then find
the area using the formula A = 1 ap.
2
Because the pentagon has 5 sides, m ACB = 360 = 72.
5
CA and CB are radii, so CA = CB. Therefore,
HL Theorem, so m ACM = 1 m ACB = 36.
ACM
BCM by the
2
HELP
GEOMETRY
Trigonometry and Area
LESSON 10-5
Additional Examples
(continued)
Use the cosine ratio
to find a.
cos 36° =
a
18
a = 18(cos 36°)
Use the sine ratio
to find AM.
Use the ratio.
Solve.
Use AM to find p. Because ACM
pentagon is regular, p = 5 • AB.
sin 36° =
AM
18
AM = 18(sin 36°)
BCM, AB = 2 • AM. Because the
So p = 5 • (2 • AM) = 10 • AM = 10 • 18(sin 36°) = 180(sin 36°).
HELP
GEOMETRY
Trigonometry and Area
LESSON 10-5
Additional Examples
(continued)
Finally, substitute into the area formula A = 1 ap.
2
A=
1
• 18(cos 36°) • 180(sin 36°)
2
Substitute for a and p.
A = 1620(cos 36°) • (sin 36°)
Simplify.
A
Use a calculator.
The area of the garden is about 770 ft2.
Quick Check
HELP
GEOMETRY
Trigonometry and Area
LESSON 10-5
Additional Examples
A triangular park has two sides that measure 200 ft and 300 ft
and form a 65° angle. Find the area of the park to the nearest hundred
square feet.
Use Theorem 10-8: The area of a triangle is one
half the product of the lengths of two sides and the
sine of the included angle.
Area = 1 • side length • side length
2
Theorem 10-8
• sine of included angle
Area = 1 • 200 • 300 • sin 65°
Substitute.
Area = 30,000 sin 65°
Simplify.
2
Use a calculator
The area of the park is approximately 27,200 ft2.
HELP
Quick Check
GEOMETRY