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'Circle in a Semicircle' printed from http://nrich.maths.org/

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The diagram shows a semi-circle containing a circle which touches the circumference of the semicircle and goes through its centre. What fraction of the semicircle is shaded?

Semi Circle containing a circle

 

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.