Try this beautiful problem from Geometry based on Ratio of the area of square and circle.
A circle with radius 1 is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?
Geometry
Circle
Square
Answer:$\frac{1}{2}$
AMC-8 (2011) Problem 25
Pre College Mathematics
Join the diagonals of the smaller square (i.e GEHF)
Can you now finish the problem ..........
The circle's shaded area is the area of the smaller square(i.e. GEHF) subtracted from the area of the circle
and The area between the two squares is Area of the square ABCD - Area of the square EFGH
can you finish the problem........
Given that the Radius of the circle with centre O is 1.Therefore The area of the circle is \(\pi (1)^2\)=\(\pi\) sq.unit
The diameter of the circle is 2 i.e \(EF=BC=2\) unit
The area of the big square i.e \(ABCD=2^2=4\) sq.unit
\(OE=OH=1\) i.e \(EH=\sqrt{(1^2+1^2)}=\sqrt 2\)
Therefore the area of the smaller square is \((\sqrt 2)^2=2\)
The circle's shaded area is the area of the smaller square(i.e. GEHF) subtracted from the area of the circle =\(\pi\) - 2
The area between the two squares is Area of the square ABCD - Area of the square EFGH=4-2=2 sq.unit
The ratio of the circle's shaded area to the area between the two squares is \( \frac{\pi - 2}{2} \approx \frac{3.14-2}{2} = \frac{1.14}{2} \approx \frac{1}{2}\)
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