10-5 Trigonometry and Area Lesson GQ: How can the trigonometric ratios to find the area of a polygon?

Trigonometry and Area Find the area of the regular pentagon with 8cm sides. A = ½ (a)(40) 8 cm To find a: Find the measure of < XCZ: 360/5 = 72 C Find the measure of < XCY: 72/2 = 36 36 3. Use the triangle and trig to solve for a: a o = tan 36 = 4 a a a = 4 tan 36 a = 5.51 X 4 cm Z Use a to solve for area: A = ½ (5.51)(40) Y Area: 110.2 cm2

Find the area of a regular octagon with a perimeter of 80 in Find the area of a regular octagon with a perimeter of 80 in. Give the area to the nearest tenth. To find the angle measure: 360/8 = 45 45/2 = 22.5 10 in o = tan 22.5 = 5 a a a = 5 tan 22.5 a = 12.07 a Use formula: A = ½ (12.07)(80) 5 in A = 482.8 in2

The Castel del Monte, built on a hill in southern Italy circa 1240, makes extraordinary use of regular octagons. One regular octagon, the inner courtyard, has a radius of 16 m. Find the area of the courtyard.

The Castel del Monte, built on a hill in southern Italy circa 1240, makes extraordinary use of regular octagons. One regular octagon, the inner courtyard, has a radius of 16 m. Find the area of the courtyard. Measure of central angle: 360/8 = 45 then, 45/2 = 22.5 16 a To solve for a: cos 22.5 = a 16 a = (cos 22.5)(16) x a = 14.78 To solve for x: sin 22.5 = x 16 x = (sin 22.5)(16) x = 6.12 Use formula to solve for Area: A = ½ (14.78)(97.92) A = 723.63 m2

Finding the Area of a Triangle Area of Triangle ABC = ½ bh You can find height by using the sin A: sin A = h c B h = c(sin A) a c h So, replace h with c(sin A): Area = ½ bc(sin A) A C b *Use if you have SAS (a known angle enclosed by two known sides.

Finding the Area of a Triangle Area of Triangle ABC = ½ bh Area = ½ bc(sin A) Area = ½ (52)(24)(sin 76) B Area = 605.46 in2 a 24 in h 76 A C 52 in

The surveyed lengths of two adjacent sides of a triangular plot of land are 412 ft and 386 ft. The angle between the sides is 71. Find the area of the plot. A = ½ bc(sin A) A = ½ (412)(386)(sin 71) 412 ft A = 75183.86 ft2 71 386 ft

Two sides of a triangular building plot are 120 ft and 85ft long Two sides of a triangular building plot are 120 ft and 85ft long. They include an angle of 85. Find the area of the building plot to the nearest square foot. A = ½ bc(sin A) 85 120 ft 85 ft A = ½ (120)(85)(sin 85) A = 5081 ft2