Two squares and two equilateral triangles. What fraction is hatched?
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Half of the large square is hatched.
An algebraic approach uses the fact that the height of an equilateral triangle is √3/2 times the base. Setting the base to 1, you can see that the diagonal of the large square is two heights and one base, so its side x=(1+√3)/√2.
The hatched area is 1+√3/2. This is exactly half of the total area x2=2+√3.