Arcs in Boundaries

Geometry Level 3

A semi-circle (arc #1) is inscribed in the small rectangle, about which the medium-sized semi-circle (arc #2) is circumscribed.

It is then again inscribed in the large rectangle, about which the largest semi-circle (arc #3) is again circumscribed.

The ratio of the area between arcs #1 and #2 to the area between arcs #2 and #3 is __________ . \text{\_\_\_\_\_\_\_\_\_\_}.

1 : 3 1 : 2 3 : 5 1 : 1

Rupen Kohli by Rupen Kohli , 36, India

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1 solution

The dimensions of a rectangle is R R and 2 R 2R , where R R is the radius of the semi-circle inscribed by the rectangle. Then, the semi-circle than inscribes this rectangle will have the radius R 2 R\sqrt2 .

So if the smallest semi-circle has the radius r r , the radius of the next semi-circle is r 2 r\sqrt2 , and the radius of the largest circle in the figure is 2 r 2r . So now the ratio of the annular areas is:

π ( r 2 ) − π ( 2 r 2 ) π ( 2 r 2 ) − π ( 4 r 2 ) \frac{\pi(r^2) - \pi(2r^2)}{\pi(2r^2) - \pi(4r^2)} = 1 / 2 =1/2

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