# Four Squares and a Semi-Circle +-- {.image} [[FourSquaresandaSemiCircle.png:pic]] > If the biggest square has area $18$, what's the total area of all four squares? =-- ## Solution by [[Intersecting Chords]] Let $a$, $b$, $c$, and $d$ be the side lengths of the four squares in ascending order. Then $d = a + c$. Reflecting the diagram in the diameter produces the following diagram, in which $A P$ is the diagonal of the smallest square, $P B$ is the reflection of the diagonal of the second largest, and $P D$ is the reflection of the side of the second smallest. +-- {.image} [[FourSquaresandaSemiCircleLabelled.png:pic]] =-- In this diagram, $A P$ has length $\sqrt{2} a$, $P C$ and $P D$ have length $b$, and $P B$ has length $\sqrt{2} c$. Using the [[intersecting chords theorem]], then, these lengths fit into the equation $$ b^2 = (\sqrt{2} a) \times (\sqrt{2} c) = 2 a c $$ Now, since $d = a + c$, the area of the largest square is equal to: $$ d^2 = (a + c)^2 = a^2 + 2 a c + c^2 = a^2 + b^2 + c^2 $$ Therefore, the three smaller squares have the same combined area as the largest. The total area is thus $36$.