9.5 – Trigonometry and Area

Area of a Regular Polygon Remember from Chapter 7 Theorem 7-12: The area of a regular polygon is half the product of the apothem and the perimeter. a p A = 1/2 a p Note that the perimeter (p) is equal to the length of a side (s) times the number of sides (n). p = n • s

Find the area of the regular pentagon. Example 1 Find the area of the regular pentagon. First find the perimeter. p = n • s p = (5) (10) p = 50 ft Now find the area. A = ½ a p A = ½ (8) (50) A = 200 ft2

Using Trigonometry to Find Area In the previous example, you were given both the apothem and the length of one side of the regular pentagon. What if you only know one of the measurements and are asked to calculate the area of the polygon? You can still calculate the area using your trigonometric functions.

Example 2 Find the area of a regular decagon with the length of one side equal to 12 cm. We know the formula to find the area. A = 1/2 a p And we can easily find the perimeter. p = n • s p = 10 • 12 p = 120 cm. But how can we find the apothem?

Example 2 (continued) You must first remember from Chapter 7 that the vertex angle of the isosceles triangle formed by the radii of a regular polygon has a measure equal to: vertex angle radius apothem

Example 2 (continued) Remember also that the apothem bisects the vertex angle of the isosceles triangle formed by the radii of a regular polygon. The apothem also bisects the side of the polygon.

Example 2 (continued) Find the area of a regular decagon with the length of one side equal to 12 cm. Now find the apothem using the tangent function. And finally, calculate the area. A = ½ a p A = ½ (18.47) (120) A = 1,108 cm2

Example 3 Find the area of a regular hexagon with a perimeter of 90 ft. Give your answer to the nearest foot. A B C D a p = 90 ft. Find the length of one side and segment AD: p = n • s 90 = 6 • s 15 ft = s Find the measure of the vertex angle?

Example 3 (continued) A B C D a Find the area of a regular hexagon with a perimeter of 90 ft. Give your answer to the nearest foot. Next, find the apothem using the tangent function. Finally, calculate the area. Note: Since this is a 30°-60°-90° triangle, the apothem (long leg) is equal to 3 • 7.5 cm (short leg). A = ½ a p A = ½ (13.0) (90) A = 585 ft2

Example 4 Find the area of a regular pentagon with a radius of 12 m. B C D a Find the area of a regular pentagon with a radius of 12 m. Vertex Angle: Apothem (Long Leg):

Example 4 (continued) 12 A B C D a Find the area of a regular pentagon with a radius of 12 m. Length of one side of Pentagon: Perimeter: p = n • s = 5 • 14.1 = 70.5 m Area: A = ½ a p = ½ (9.7) (70.5) = 342 m2

Theorem 9-1: Area of a Triangle Given SAS The area of a triangle is one half the product of the lengths of two sides and the sine of the included angle. C A B c a b

Example 5 Find the area of a each triangle. b. a.

Homework: p. 500 1-17, 20, 24, 26