# 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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