1. Using Trig Functions to Find the Areas of Regular Polygons
10.5 Geometry
Using Trig Functions to Find the Areas 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
April 4, 2019

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
April 4, 2019

4. Non-Special Triangles
Find the area of a regular octagon if the length of the sides is 10.
April 4, 2019

5. Step 1
Draw a regular octagon with side length 10. 10
April 4, 2019

6. Step 2
Locate the center and draw a central angle. 10
April 4, 2019

7. Step 3 Determine the measure of the central angle. 10 45
April 4, 2019

8. Step 4
Draw the apothem. 10
45
April 4, 2019

9. Step 5
The apothem bisects the angle and the side. Write their measures. 10
22.5
45 5
April 4, 2019

10. Step 6 Use a trig function to find the apothem. 10 22.5 a 5
April 4, 2019

11. Step 7
Find the perimeter.
p = 10  8
p = 80 10
12.07
April 4, 2019

12. Step 8
Find the area.
p = 80
A = 482.8 10
12.07
April 4, 2019

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
April 4, 2019

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
April 4, 2019

15. Another example Solve the equation: 36 6 x
Use a scientific calculator or use the table on page 845.
April 4, 2019

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
April 4, 2019

17. Another example
The area is:
36
8.72 6 x
April 4, 2019