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Geometry
Using Trig Functions to Find the
Areas of Regular Polygons
Goals



July 6, 2017
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
Finding Internal Angles
Find the area of the regular pentagon.
Where did 36 come from?
36
6
July 6, 2017
360
Each central angle
measures 1/5 of 360, or
72.
The apothem bisects the
central angle. Half of 72
is 36.
Non-Special Triangles
Find the area of a regular octagon
if the length of the sides is 10.
July 6, 2017
Step 1

Draw a regular octagon with side
length 10.
10
July 6, 2017
Step 2

Locate the center and draw a central
angle.
10
July 6, 2017
Step 3

Determine the measure of the
central angle.
360
 45
8
10
45
July 6, 2017
Step 4

Draw the apothem.
10
45
July 6, 2017
Step 5

The apothem bisects the angle and
the side. Write their measures.
10
22.5
45
5
July 6, 2017
Step 6

Use a trig function to find the
apothem.
10
22.5
a
5
July 6, 2017
5
tan22.5 
a
5
a
tan22.5
a  12.07
Step 7

Find the perimeter.
p = 10  8
p = 80
10
12.07
July 6, 2017
Step 8

Find the area.
1
2

1
2
ap
12.07  80 
 482.8
p = 80
A = 482.8
10
12.07
July 6, 2017
A
Another example
Find the area of the regular pentagon.
What is the apothem?
6
36
6
What is the perimeter?
Don’t know.
Let’s find it.
July 6, 2017
Another example
Find the area of the regular pentagon.
What trig function can be
used to find x?
36
(SOHCAHTOA)
6
Equation:
x
July 6, 2017
TANGENT
x
tan36 
6
Another example
Solve the equation:
tan36 
36
6
6 tan36  x
6(.7265)  x
x
Use a scientific calculator or
use the table on page 845.
July 6, 2017
x
6
x  4.36
Another example
x = 4.36
One side of the
pentagon measures?
36
8.72
6
The perimeter is
4.36
July 6, 2017
8.72 (2  4.36)
43.59 (5  8.72)
Another example
The area is:
36
8.72
6
1
2

1
2
ap
6   43.59 
 130.78
x
July 6, 2017
A