## Four Circles

In the diagram, two of the three dark shaded circles have diameters of $4$ units, and the third has a diameter of $2$. What must be the length of the diameter of the white shaded circle to make the sum of the dark shaded areas $A,B$, and $C$ equal to the white shaded area?

Source: NCTM Mathematics Teacher, February 2006

SOLUTION

area of $A$ + area of $B$ + area of $C$ = area of white circle – area of $D$ – area of $E$ – area of $F$
(area of $A$ + area of $D$) + (area of $B$ + area of $E$) + (area of $C$ + area of $F$) = area of white circle
$\pi 2^2+\pi 2^2+\pi 1^2=\pi x^2$ where $x$ is the radius of the white circle
$9\pi=\pi x^2$
$x^2=9$
$x=\pm 3$
Diameter of white circle = $2\times 3=6$

Answer: $6$ units