# Solution to the Two Overlapping Squares Puzzle +-- {.image} [[TwoOverlappingSquares.png:pic]] > Two overlapping squares. What's the shaded area? =-- ## Solution by [[Pythagoras' Theorem]] and [[Similar Triangles]] +-- {.image} [[TwoOverlappingSquaresLabelled.png:pic]] =-- Let the two squares have side lengths $a$ and $b$, with $a$ for the light blue and $b$ for the dark blue. Applying [[Pythagoras' theorem]] to triangle $A B C$ shows that: $$ b^2 = a^2 + 5^2 $$ Triangle $A E D$ is [[similar]] to triangle $A B C$ with scale factor $\frac{a}{b}$, so the length of $A E$ is $\frac{a^2}{b}$. The area of the shaded region is then given by: $$ b \times \left(b - \frac{a^2}{b}\right) = b^2 - a^2 = 5^2 = 25 $$