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10.6Areas & TrigonometrySometimes we need to divide a figure into triangles to find the area. When we do this, at times we may need to use trigonometry to find the needed lengths. A barn is in the shape of a regular octagon with a 56 diameter. Find the area of the barn floor. We know that we can divide the octagon into 8 congruent isosceles triangles. The vertex angle is 45 degrees. (360 divided by 8 = 45).The base angle is (180 * (8-2)) = 1080. Now 1080 divide by 8 = 135 degrees. Finally, divide 135 by 2 = 67.5 degrees for the base angles. Now use trigonometry to find the We need to divide the diameter in half to find the radius of each side.56 divided by 2 = 28.Now use trigonometry to find the height and base of each angle. Sin 67.5 = a a = 26. 28Next use either the Pythagorean Theoremor cosine to find ½ the base:262+ b2= 282b = 10.7. Now multiply by 2 and get the base of 21.4Find the perimeter 8 * 21.4 = 171 ft.Finally A = ½ ap so A = 1.2 (26)(171) A = 2223 ft2
Find thearea of a parallelogram:We will use trig to find the height. Remember, the height is perpendicular to the base. Opposite angles of a parallelogram are congruent alsoSin 24 = hh = 5.2913 Now A = bh A = 27 * 5.29 = 142.8 un2The pentagon base can be divided into 5 congruent triangles with vertex angle of 72 degrees and base angles of 54 degrees.We get the base angle by taking the number of degrees ina pentagon [180 *(number of sides –2)], so 180 * 3 = 540. Since this is a regular polygon, divide 540 by 5 = 108. Now we divide the angle by 2 = 54.The vertex angle: There are 360 degrees in a circle. Now divide 360 by 5 = 72.24º1327