Circle in a semicircle
Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?
Problem
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?
Image
![Circle in a Semicircle Circle in a Semicircle](/sites/default/files/styles/large/public/thumbnails/content-04-weekly-prob6-1-1.png?itok=2IOF29dN)
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: $\frac12$
Image
![Circle in a Semicircle Circle in a Semicircle](/sites/default/files/styles/large/public/thumbnails/content-04-weekly-prob6-circle%252520in%252520a%252520semicircle%2525201.png?itok=WDw_Y5ff)
Image
![Circle in a Semicircle Circle in a Semicircle](/sites/default/files/styles/large/public/thumbnails/content-04-weekly-prob6-circle%252520in%252520a%252520semicircle%2525202.png?itok=kezBd0Hm)
Image
![Circle in a Semicircle Circle in a Semicircle](/sites/default/files/styles/large/public/thumbnails/content-04-weekly-prob6-circle%252520in%252520a%252520semicircle%2525203.png?itok=jcPiH69J)
Image
![Circle in a Semicircle Circle in a Semicircle](/sites/default/files/styles/large/public/thumbnails/content-04-weekly-prob6-circle%252520in%252520a%252520semicircle%2525204.png?itok=mmZFOL1n)