Annulus area
Weekly Problem 38 - 2011
Given three concentric circles, shade in the annulus formed by the smaller two. What percentage of the larger circle is now shaded?
Given three concentric circles, shade in the annulus formed by the smaller two. What percentage of the larger circle is now shaded?
Age
14 to 16
Challenge level
Problem
The three circles in the diagram have the same centre and have radii 3cm, 4cm and 5cm. What percentage of the larger circle is shaded?
Image
Student Solutions
Ans: 28%
The area of the larger circle is 25$ \pi $ cm $^2$.
The area of the smallest circle is 9$ \pi $ cm $^2$.
The area of the middle circle is 16$ \pi $ cm $ ^2 $.
Therefore the area of the ring is (16$ \pi $ - 9 $ \pi $) cm $ ^2 $ i.e. 7 $ \pi $ cm $^2 $.
Therefore the fraction shaded is $\frac{7 \pi }{25\pi }$
Therefore the percentage shaded is 28%