Geometry
Using Trig Functions to Find the
Areas of Regular Polygons
Goals



April 12, 2015
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
April 12, 2015
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.
April 12, 2015
Step 1

Draw a regular octagon with side
length 10.
10
April 12, 2015
Step 2

Locate the center and draw a central
angle.
10
April 12, 2015
Step 3

Determine the measure of the
central angle.
360
 45
8
10
45
April 12, 2015
Step 4

Draw the apothem.
10
45
April 12, 2015
Step 5

The apothem bisects the angle and
the side. Write their measures.
10
22.5
45
5
April 12, 2015
Step 6

Use a trig function to find the
apothem.
10
22.5
a
5
April 12, 2015
5
tan22.5 
a
5
a
tan22.5
a  12.07
Step 7

Find the perimeter.
p = 10  8
p = 80
10
12.07
April 12, 2015
Step 8

Find the area.
1
2

1
2
ap
12.07 80 
 482.8
p = 80
A = 482.8
10
12.07
April 12, 2015
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.
April 12, 2015
Another example
Find the area of the regular pentagon.
What trig function can be
used to find x?
36
(SOHCAHTOA)
6
Equation:
x
April 12, 2015
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.
April 12, 2015
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
April 12, 2015
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
April 12, 2015
A