Transcript Finding Areas Using Trig
Geometry
Using Trig Functions to Find the Areas of Regular Polygons
Goals Determine the central angle of a polygon.
Find the area of polygons not comprised of 30-60-90 or 45-45-90 triangles Use trig functions to find the apothem and the length of a side of a polygon April 26, 2020
Finding Internal Angles Find the area of the regular pentagon.
6 Where did 36 come from?
36 360 Each central angle measures 1/5 of 360 , or 72 .
The apothem bisects the central angle. Half of 72 is 36 .
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Non-Special Triangles Find the area of a regular octagon if the length of the sides is 10.
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Step 1 Draw a regular octagon with side length 10.
10 April 26, 2020
Step 2 Locate the center and draw a central angle.
10 April 26, 2020
Step 3 Determine the measure of the central angle.
360 8 45 10 45 April 26, 2020
Step 4 Draw the apothem.
10 45 April 26, 2020
Step 5 The apothem bisects the angle and the side. Write their measures.
April 26, 2020 10 22.5
45 5
Step 6 Use a trig function to find the apothem.
10 22.5
tan22.5
a a
5
a
5 tan22.5
12.07
a 5 April 26, 2020
Step 7 Find the perimeter.
p = 10 8 p = 80 10 12.07
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April 26, 2020 Step 8 Find the area.
p = 80 A = 482.8
A
1 2 1 2
ap
482.8
10 12.07
Another example Find the area of the regular pentagon.
6 36 What is the apothem?
6 What is the perimeter?
Don’t know.
Let’s find it.
April 26, 2020
Another example Find the area of the regular pentagon.
6 36 x What trig function can be used to find x?
TANGENT (SOHCAH TOA ) Equation: tan36
x
6 April 26, 2020
Another example April 26, 2020 Solve the equation:
tan36
6 36 x
6 tan36 6(.7265)
x
Use a scientific calculator or use the table on page 845.
x
6
x x
4.36
8.72
Another example 6 36 4.36
x = 4.36
One side of the pentagon measures?
8.72
(2 4.36) The perimeter is 43.59
(5 8.72) April 26, 2020
8.72
Another example 6 36 x The area is:
A
1 2
ap
1 2
130.78
April 26, 2020