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Singapore Math Olympiad

Problem on Area of Circle | SMO, 2010 (Junior) | Problem 29

Try this beautiful problem on area of circle from SMO, Singapore Mathematics Olympiad, 2010. You may use sequential hints to solve the problem.

Try this beautiful problem on area of circle from SMO, Singapore Mathematics Olympiad, 2010.

Problem – Area of Circle (SMO Test)


Let ABCD be a rectangle with AB = 10 . Draw circles \(C_1 \) and \(C_2\) with diameters AB and CD respectively. Let P,Q be the intersection points of \(C_1\) and \(C_2\) . If the circle with diameter PQ is tangent to AB and CD , then what is the area of the shaded region ?

Problem on Area of Circle from SMO
  • 25
  • 20
  • 22
  • 23

Key Concepts


Area of Circle

2D – Geometry

Area of Rectangle

Check the Answer


But try the problem first…

Answer: 25

Source
Suggested Reading

Singapore Mathematics Olympiad

Challenges and thrills – Pre – college Mathematics

Try with Hints


First Hint………………………

If you are really got stuck with this sum then we can start from here:

The diagram will be like this . So let ‘N’ be the midpoint of CD .

so \(\angle {PNQ} = 90^\circ\)

so PQ = 5 \(\sqrt {2}\)

Second Hint …………………………….

Now let us try to find the area of the shaded region

A = \(2 [ \frac {1}{2}\pi (\frac {PQ)}{2})^2 + \frac {1}{2}(PN)^2 – \frac {1}{4} \pi (PN)^2]\)

= \(2 [\frac {1}{2}\pi (\frac {5\sqrt{2}}{2})^2+\frac {1}{2}.5^2 – \frac {1}{4}\pi . 5^2]\)

= 25

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