The area of the semicircle in Figure A is half the area of the circle in Figure B. The area of a square inscribed in the semicircle, as shown, is what fraction of the area of a square inscribed in the circle? Express your answer as a common fraction.




Guest Jun 30, 2018

Call the radius  of the semi-circle and the circle, r

Call the side of the  small square, s


So...using the Pythagorean Theorem, we have that


(1/2)s^2  + s^2  = r^2

(1/2)s^2 + s^2  = r^2

(5/4)s^2  = r^2

√5/2 s  = r

s  =   2r / √5



So...the area of the small square = s^2  =   (4/5)r^2


And..in the larger circle....the side of the square is √2r

So....the area of the square in the larger circle is just    2r^2


So....the  area of the smaller square's area to  the larger square's   area is


(4/5)r^2  / 2 r^2   =


4/10  = 


2/5  of the large square's area



cool cool cool

CPhill  Jun 30, 2018

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