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solid mensuration

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Find the area of a portion of a semicircle of radius 15 m that is outside the area of an inscribed square if the base of the square lies on the diameter of the semicircle.

Please include solution. Thanks.

  • solid mensuration -

    make a sketch, putting the complete figure on the x-y grid
    draw a line from the centre of the semicircle to the vertex of the square , calling that point P(x,y)

    so the base of the square is 2x and its height is y
    so clearly
    y = 2x

    also x^2 + y^2 = 15^2
    x^2 + (2x)^2 = 225
    5x^2 = 225
    x^2 = 45
    x = 3√5
    y = 6√5
    area of square = 2xy
    = 2(3√5)(6√5) = 180

    area of semicircle = (1/2)π(15)^2 = 225π/2

    area between square and semicircle
    = 225π/2 - 180 or appr 173.43

    check my arithmetic

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