4. In the diagram, the smaller circle touches the larger circle and also passes through its centre.
What fraction of the area of the larger circle is outside the smaller circle?
A $\frac{2}{3}$
B $\frac{3}{4}$
C $\frac{4}{5}$
D $\frac{5}{6}$
E $\frac{6}{7}$
SolUtion
B

Because the smaller circle touches the larger circle and passes through its centre, the diameter of the smaller circle is half that of the larger circle. It follows that the radius of the smaller circle is half that of the larger circle.
The area of a circle with radius $r$ is $\pi r^{2}$. Therefore, the area of a circle with radius $\frac{1}{2} r$ is $\pi\left(\frac{1}{2} r\right)^{2}=\frac{1}{4} \pi r^{2}$. It follows that the area of the smaller circle is $\frac{1}{4}$ of the area of the larger circle. Therefore $\frac{1}{4}$ of the area of the larger circle is inside the smaller circle. Hence $\frac{3}{4}$ of the area of the larger circle is outside the smaller circle.