1. Geometry Using Trig Functions to Find the Areas of Regular Polygons

2. November 2, 2015 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

3. November 2, 2015 Finding Internal Angles 6 36 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. 360

4. November 2, 2015 Non-Special Triangles Find the area of a regular octagon if the length of the sides is 10.

5. November 2, 2015 Step 1  Draw a regular octagon with side length 10. 10

6. November 2, 2015 Step 2  Locate the center and draw a central angle. 10

7. November 2, 2015 Step 3  Determine the measure of the central angle. 10 45

8. November 2, 2015 Step 4  Draw the apothem. 10 45

9. November 2, 2015 Step 5  The apothem bisects the angle and the side. Write their measures. 10 45 22.5 5

10. November 2, 2015 Step 6  Use a trig function to find the apothem. 10 22.5 5 a

11. November 2, 2015 Step 7  Find the perimeter. 10 12.07 p = 10  8 p = 80

12. November 2, 2015 Step 8  Find the area. 10 12.07 p = 80 A = 482.8

13. November 2, 2015 Another example 6 36 Find the area of the regular pentagon. What is the apothem? 6 What is the perimeter? Don’t know. Let’s find it.

14. November 2, 2015 Another example 6 36 Find the area of the regular pentagon. What trig function can be used to find x? TANGENT (SOHCAHTOA) Equation: x

15. November 2, 2015 Another example 6 36 Solve the equation: x Use a scientific calculator or use the table on page 845.

16. November 2, 2015 Another example 6 36 x = 4.36 One side of the pentagon measures? 8.72(2  4.36) The perimeter is 43.59 (5  8.72) 4.36 8.72

17. November 2, 2015 Another example 6 36 x The area is: 8.72