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What is the radius of circle if the pink area equals the green one?

  1. flysky profile image84
    flyskyposted 5 years ago

    What is the radius of circle if the pink area equals the green one?

    https://usercontent1.hubstatic.com/7724798_f260.jpg

  2. dezalyx profile image94
    dezalyxposted 5 years ago

    Because the triangle is equilateral, the angle for the pink area is 60 degrees, which is 1/6 of the 360 degree circle. So the area of the pink shaded region is pi * r^2 / 6.

    The height of the triangle is sqrt (4 - 1) = sqrt (3). Which makes the area of the whole triangle= sqrt (3) * 2 / 2 = sqrt (3). So the area of the green shaded region = area of the whole triangle - area of the pink shaded region = sqrt (3) - pi * r^2 / 6.

    Given that the area of the pink shaded region = area of the green region, we get the following equation:
    pi * r^2 / 6 = sqrt (3) - pi * r^2 / 6

    Solving for r:
    pi * r^2 / 3 = sqrt (3)
    r^2 = 3 * sqrt (3) / pi
    r = sqrt [3 * sqrt (3) / pi]

  3. GreenMathDr profile image74
    GreenMathDrposted 5 years ago

    Since the area of the triangle is sq rt 3, then pi r^2/6 must equal (rad 3)/2. cross multiplying and dividing by 3 and simplifying gives approximately 1.654 = r^2 taking the sq rt of each side gives approximately 1.286.

  4. flysky profile image84
    flyskyposted 5 years ago

    Both solutions are correct! (so I select the first answer as the best) The area of triangle is sqrt(3), and than we use formula for the pink area: A=r^2 *pi * alpha /360, where the angle alpha is 60 (since we have equilateral triangle).

 
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