Using Trig to find area 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 December 3, 2018

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 December 3, 2018

Non-Special Triangles Find the area of a regular octagon if the length of the sides is 10. December 3, 2018

Step 1 Draw a regular octagon with side length 10. 10 December 3, 2018

Step 2 Locate the center and draw a central angle. 10 December 3, 2018

Step 3 Determine the measure of the central angle. 10 45 December 3, 2018

Step 4 Draw the apothem. 10 45 December 3, 2018

Step 5 The apothem bisects the angle and the side. Write their measures. 10 22.5 45 5 December 3, 2018

Step 6 Use a trig function to find the apothem. 10 22.5 a 5 December 3, 2018

Step 7 Find the perimeter. p = 10  8 p = 80 10 12.07 December 3, 2018

Step 8 Find the area. p = 80 A = 482.8 10 12.07 December 3, 2018

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

Another example Find the area of the regular pentagon. What trig function can be used to find x? TANGENT (SOHCAHTOA) Equation: 36 6 x December 3, 2018

Another example Solve the equation: 36 6 x Use a scientific calculator or use the table on page 845. December 3, 2018

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 December 3, 2018

Another example The area is: 36 8.72 6 x December 3, 2018

Use trigonometry to find a. Trigonometry and Area LESSON 10-5 Additional Examples Find the area of a regular polygon with 10 sides and side length 12 cm. Find the perimeter p and apothem a, and then find the area using the formula A = ap. 1 2 Because the polygon has 10 sides and each side is 12 cm long, p = 10 • 12 = 120 cm. Use trigonometry to find a. 360 10 Because the polygon has 10 sides, m ACB = = 36. CA CB 1 2 and are radii, so CA = CB. Therefore, ACM BCM by the HL Theorem, so m ACM = m ACB = 18 and AM = AB = 6.

Now substitute into the area formula. Trigonometry and Area LESSON 10-5 Additional Examples (continued) tan 18° = 6 a Use the tangent ratio. a = 6 tan 18° Solve for a. Now substitute into the area formula. A = ap 1 2 A = • • 120 1 2 6 . tan 18° Substitute for a and p. A = 360 tan 18° Simplify. 360 18 Use a calculator. The area is about 1108 cm2. Quick Check

Because the pentagon has 5 sides, m ACB = = 72. Trigonometry and Area LESSON 10-5 Additional Examples The radius of a garden in the shape of a regular pentagon is 18 feet. Find the area of the garden. Find the perimeter p and apothem a, and then find the area using the formula A = ap. 1 2 Because the pentagon has 5 sides, m ACB = = 72. CA and CB are radii, so CA = CB. Therefore, ACM BCM by the HL Theorem, so m ACM = m ACB = 36. 360 5 1 2

Use the cosine ratio to find a. Use the sine ratio to find AM. Trigonometry and Area LESSON 10-5 Additional Examples (continued) Use the cosine ratio to find a. Use the sine ratio to find AM. a = 18(cos 36°) AM = 18(sin 36°) Use the ratio. Solve. cos 36° = a 18 sin 36° = AM Use AM to find p. Because ACM BCM, AB = 2 • AM. Because the pentagon is regular, p = 5 • AB. So p = 5 • (2 • AM) = 10 • AM = 10 • 18(sin 36°) = 180(sin 36°).

Finally, substitute into the area formula A = ap. Trigonometry and Area LESSON 10-5 Additional Examples (continued) 1 2 Finally, substitute into the area formula A = ap. A = • 18(cos 36°) • 180(sin 36°) 1 2 Substitute for a and p. A = 1620(cos 36°) • (sin 36°) Simplify. A Use a calculator. The area of the garden is about 770 ft2. Quick Check

Area = • side length • side length Theorem 10-8 Trigonometry and Area LESSON 10-5 Additional Examples A triangular park has two sides that measure 200 ft and 300 ft and form a 65° angle. Find the area of the park to the nearest hundred square feet. Use Theorem 10-8: The area of a triangle is one half the product of the lengths of two sides and the sine of the included angle. Area = • side length • side length 1 2 Theorem 10-8 • sine of included angle Area = • 200 • 300 • sin 65° 1 2 Substitute. Area = 30,000 sin 65° Simplify. Use a calculator Quick Check The area of the park is approximately 27,200 ft2.