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Math Help - Finding the perimeter and area of the shaded portion

  1. #1
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    Finding the perimeter and area of the shaded portion

    Hi, I was wondering if I could get some help with this problem... After doing the area I got 9.78" and I don't know how to do the perimeter. If someone could go over this with me, that would be great!
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  2. #2
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    Re: Finding the perimeter and area of the shaded portion

    Have a look at this webpage.
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  3. #3
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    Re: Finding the perimeter and area of the shaded portion

    To find the area of the shaded region:
    1) Find the area of the circular sector. This circle has radius 8 and so area \pi (8)^2= 64\pi. the sector is \frac{72}{360}= \frac{1}{5} of the circle and so has area \frac{64}{5}\pi.
    2) Find the area of the triangle. It has center angle of 72 degrees so, by the cosine law, letting c be the opposite side of the triangle, [tex]c= 64+ 64- 2(64)cos(72)= 128(1- cos(72)). Dropping a perpendicular to the opposite side, we have two right triangles with hypotenuse of length and angle 72/2= 36 degrees. The altitude of the triangle is a "near side" of those right triangles and so has length 8cos(36). The area of a triangle is (1/2)base times height which, here, is [tex](1/2)(128(1- cos(72))(8 cos(36))= 512cos(36)(1- cos(72)).
    3. Subtract the area of the triangle from the area of the sector.

    To find the perimeter add the length of the base of that third side of the triangle, given above, to the length of the arc, which is 1/5 the circumference of the circle.
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