2008 AMC 12B Problems/Problem 15
Problem
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let 
be the region formed by the union of the square and all the triangles, and 
be the smallest convex polygon that contains . What is the area of the region that is inside but outside ?
Solution
![[asy] real a = 1/2, b = sqrt(3)/2; draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,0)--(a,-b)--(1,0)--(1+b,a)--(1,1)--(a,1+b)--(0,1)--(-b,a)--(0,0)); draw((0,0)--(-1+a,-b)--(1+a,-b)--(1,0)--(1+b,-1+a)--(1+b,1+a)--(1,1)--(1+a,1+b)--(-1+a,1+b)--(0,1)--(-b,1+a)--(-b,-1+a)--(0,0)); filldraw((1+a,-b)--(1,0)--(1+b,-1+a)--cycle,gray(0.9)); filldraw((1+b,1+a)--(1,1)--(1+a,1+b)--cycle,gray(0.9)); filldraw((-1+a,1+b)--(0,1)--(-b,1+a)--cycle,gray(0.9)); filldraw((-b,-1+a)--(0,0)--(-1+a,-b)--cycle,gray(0.9)); [/asy]](http://latex.artofproblemsolving.com/8/c/7/8c7af6121e2831aa5bf8db7a271830fc340d0c5d.png)
The equilateral triangles form trapezoids with side lengths 
(half a unit hexagon) on each face of the unit square. The four triangles "in between" these trapezoids are isosceles triangles with two side lengths 
and an angle of 
in between them, so the total area of these triangles (which is the area of $S \minus{} R$ (Error compiling LaTeX. Unknown error_msg)) is, by the Law of Sines, 
which makes the answer 
.
See Also
Region with Squares and Triangles
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