AMC12 2008 12B, Problem 13

Vertex $E$ of equilateral $\triangle{ABE}$ is in the interior of unit square $ABCD$. Let $R$ be the region consisting of all points inside $ABCD$ and outside $\triangle{ABE}$ whose distance from $AD$ is between $\frac{1}{3}$ and $\frac{2}{3}$. What is the area of $R$?

Solution: The region is the shaded area: [asy] pair A,B,C,D,E; A=(0,1); B=(1,1); C=(1,0); D=(0,0); E=(1/2,1-sqrt(3)/2); draw(A--B--C--D--cycle); label("A",A,NW); dot(A); label("B",B,NE); dot(B); label("C",C,SE); dot(C); label("D",D,SW); dot(D); draw(A--E--B--cycle); label("E",E,S); dot(E); draw((1/3,0)--(1/3,1)); draw((2/3,0)--(2/3,1)); fill((1/3,0)--(1/3,1-sqrt(3)/3)--E--(2/3,1-sqrt(3)/3)--(2/3,0)--cycle,Black);[/asy] We can find the area of the shaded region by subtracting the pentagon from the middle third of the square.  The area of the middle third of the square is $\left(\frac13\right)(1)=\frac13$.  The pentagon can be split into a rectangle and an equilateral triangle. The base of the equilateral triangle is $\frac13$ and the height is $\left(\frac13\right)\left(\frac12\right)(\sqrt{3})=\frac{\sqrt{3}}{6}$.  Thus, the area is $\left(\frac{\sqrt3}{6}\right)\left(\frac13\right)\left(\frac12\right)=\frac{\sqrt3}{36}$. The base of the rectangle is $\frac13$ and the height is the height of the equilateral triangle minus the height of the smaller equilateral triangle.  This is: $\frac{\sqrt3}{2}-\frac{\sqrt3}{6}=\frac{\sqrt3}{3}$ Therefore, the area of the shaded region is $\frac13-\frac{\sqrt3}{9}-\frac{\sqrt3}{36}=\fbox{\frac{12-5\sqrt{3}}{36}}.$

Answer: \frac{12-5\sqrt{3}}{36}