1. Using Trig to find area of Regular Polygons

2. Goals Determine the central angle of a polygon.
Find the area of polygons not comprised of or triangles
Use trig functions to find the apothem and the length of a side of a polygon
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3. Finding Internal Angles
Find the area of the regular pentagon.
Where did 36 come from?
Each central angle measures 1/5 of 360, or 72.
The apothem bisects the central angle. Half of 72 is 36.
36
360 6
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4. Non-Special Triangles
Find the area of a regular octagon if the length of the sides is 10.
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5. Step 1
Draw a regular octagon with side length 10. 10
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6. Step 2
Locate the center and draw a central angle. 10
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7. Step 3 Determine the measure of the central angle. 10 45
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8. Step 4
Draw the apothem. 10
45
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9. Step 5
The apothem bisects the angle and the side. Write their measures. 10
22.5
45 5
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10. Step 6 Use a trig function to find the apothem. 10 22.5 a 5
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11. Step 7
Find the perimeter.
p = 10  8
p = 80 10
12.07
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12. Step 8
Find the area.
p = 80
A = 482.8 10
12.07
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13. Another example Find the area of the regular pentagon.
What is the apothem? 6
What is the perimeter?
Don’t know.
Let’s find it.
36 6

14. Another example Find the area of the regular pentagon.
What trig function can be used to find x?
TANGENT
(SOHCAHTOA)
Equation:
36 6 x
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15. Another example Solve the equation: 36 6 x
Use a scientific calculator or use the table on page 845.
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16. Another example x = 4.36 One side of the pentagon measures?
8.72 (2  4.36)
The perimeter is
(5  8.72)
36
8.72 6
4.36
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17. Another example
The area is:
36
8.72 6 x
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18. Use trigonometry to find a.
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 = ap. 1 2
Because the polygon has 10 sides and each side is 12 cm long, p = 10 • 12 = 120 cm.
Use trigonometry to find a.
360 10
Because the polygon has 10 sides, m ACB = = 36. CA CB 1 2
and are radii, so CA = CB. Therefore, ACM BCM by the HL Theorem, so m ACM = m ACB = 18 and AM = AB = 6.

19. Now substitute into the area formula.
Trigonometry and Area
LESSON 10-5
Additional Examples
(continued)
tan 18° = 6 a
Use the tangent ratio.
a = 6
tan 18°
Solve for a.
Now substitute into the area formula.
A = ap 1 2
A = • • 120 1 2
tan 18°
Substitute for a and p.
A =
360
tan 18°
Simplify.
Use a calculator.
The area is about 1108 cm2.
Quick Check

20. Because the pentagon has 5 sides, m ACB = = 72.
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 = ap. 1 2
Because the pentagon has 5 sides, m ACB = = 72.
CA and CB are radii, so CA = CB. Therefore, ACM BCM by the
HL Theorem, so m ACM = m ACB = 36.
360 5 1 2

21. Use the cosine ratio to find a. Use the sine ratio to find AM.
Trigonometry and Area
LESSON 10-5
Additional Examples
(continued)
Use the cosine ratio to find a.
Use the sine ratio to find AM.
a = 18(cos 36°)
AM = 18(sin 36°)
Use the ratio.
Solve.
cos 36° = a 18
sin 36° = AM
Use AM to find p. Because ACM BCM, AB = 2 • AM. Because the pentagon is regular, p = 5 • AB.
So p = 5 • (2 • AM) = 10 • AM = 10 • 18(sin 36°) = 180(sin 36°).

22. Finally, substitute into the area formula A = ap.
Trigonometry and Area
LESSON 10-5
Additional Examples
(continued) 1 2
Finally, substitute into the area formula A = ap.
A = • 18(cos 36°) • 180(sin 36°) 1 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

23. Area = • side length • side length Theorem 10-8
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 = • side length • side length 1 2
Theorem 10-8
• sine of included angle
Area = • 200 • 300 • sin 65° 1 2
Substitute.
Area = 30,000 sin 65°
Simplify.
Use a calculator
Quick Check
The area of the park is approximately 27,200 ft2.