1. 10-5 Trigonometry and Area
Lesson GQ: How can the trigonometric ratios to find the area of a polygon?

2. Trigonometry and Area
Find the area of the regular pentagon with 8cm sides.
A = ½ (a)(40)
8 cm
To find a:
Find the measure of < XCZ:
360/5 = 72 C
Find the measure of < XCY:
72/2 = 36 36
3. Use the triangle and trig to solve for a: a
o = tan 36 = 4
a a
a =
tan 36
a = 5.51 X
4 cm Z
Use a to solve for area:
A = ½ (5.51)(40) Y
Area: cm2

3. Find the area of a regular octagon with a perimeter of 80 in
Find the area of a regular octagon with a perimeter of 80 in. Give the area to the nearest tenth.
To find the angle measure:
360/8 = 45
45/2 = 22.5
10 in
o = tan 22.5 = 5
a a
a =
tan 22.5
a = 12.07 a
Use formula:
A = ½ (12.07)(80)
5 in
A = in2

4. The Castel del Monte, built on a hill in southern Italy circa 1240, makes extraordinary use of regular octagons. One regular octagon, the inner courtyard, has a radius of 16 m. Find the area of the courtyard.

5. The Castel del Monte, built on a hill in southern Italy circa 1240, makes extraordinary use of regular octagons. One regular octagon, the inner courtyard, has a radius of 16 m. Find the area of the courtyard.
Measure of central angle:
360/8 = 45
then, 45/2 = 22.5 16 a
To solve for a:
cos 22.5 = a 16
a = (cos 22.5)(16) x
a = 14.78
To solve for x:
sin 22.5 = x 16
x = (sin 22.5)(16)
x = 6.12
Use formula to solve for Area:
A = ½ (14.78)(97.92)
A = m2

6. Finding the Area of a Triangle
Area of Triangle ABC = ½ bh
You can find height by using the sin A:
sin A = h c B
h = c(sin A) a c h
So, replace h with c(sin A):
Area = ½ bc(sin A) A C b
*Use if you have SAS (a known angle enclosed by two known sides.

7. Finding the Area of a Triangle
Area of Triangle ABC = ½ bh
Area = ½ bc(sin A)
Area = ½ (52)(24)(sin 76) B
Area = in2 a
24 in h 76 A C
52 in

8. The surveyed lengths of two adjacent sides of a triangular plot of land are 412 ft and 386 ft. The angle between the sides is 71. Find the area of the plot.
A = ½ bc(sin A)
A = ½ (412)(386)(sin 71)
412 ft
A = ft2 71
386 ft

9. Two sides of a triangular building plot are 120 ft and 85ft long
Two sides of a triangular building plot are 120 ft and 85ft long. They include an angle of 85. Find the area of the building plot to the nearest square foot.
A = ½ bc(sin A) 85
120 ft
85 ft
A = ½ (120)(85)(sin 85)
A = 5081 ft2