2015 UNCO Math Contest II Problems/Problem 2


[asy] filldraw(circle((0,0),3),white); filldraw(arc((-1,0),2,0,180)--cycle,grey); filldraw(arc((-2,0),1,0,180)--cycle,white); filldraw(arc((1,0),2,180,360)--cycle,grey); filldraw(arc((2,0),1,180,360)--cycle,white); [/asy]

Find the area of the shaded region. The outer circle has radius $3$. The shaded region is outlined by half circles whose radii are $1$ and $2$ and whose centers lie on the dashed diameter of the big circle.


We can translate the right shaded semicircle along the diameter of the largest circle so that the shaded region becomes a circle (with an unshaded semicircle). We then translate the right smallest semicircle along the diameter of the largest circle onto the left smallest semicircle. This creates a shaded circle of radius 2 and an unshaded circle of radius 1, whereby it is obvious that the area is $4\pi-\pi=\boxed{3\pi}$.

See also

2015 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions