Try this beautiful problem from Geometry based on Ratio of the area of square and circle.

Area of the star and circle – AMC-8, 2011 – Problem 25


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?

area of square and circle
  • $\frac{3}{2}$
  • $\frac{1}{2}$
  • $1$

Key Concepts


Geometry

Circle

Square

Check the Answer


But try the problem first…

Answer:$\frac{1}{2}$

Source
Suggested Reading

AMC-8 (2011) Problem 25

Pre College Mathematics

Try with Hints


First hint

Join the diagonals of the smaller square (i.e GEHF)

figure simplified 1

Can you now finish the problem ……….

Second Hint

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……..

Final Step

figure solution 2

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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