Using Trig Functions to Find the Areas of Regular Polygons 10.5 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 4, 2019

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

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

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

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

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

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

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

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

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

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

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

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

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

Another example x = 4.36 One side of the pentagon measures? 8.72 (2  4.36) The perimeter is 43.59 (5  8.72) 36 8.72 6 4.36 April 4, 2019

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