[[!redirects two rectangles and a square ii solution]] # Solution to the [[Six Squares in a Semi-Circle and a Quarter Circle]] Puzzle +-- {.image} [[TwoRectanglesandaSquareII.jpeg:pic]] > $6$ equal squares in a semi-circle and in a quarter circle; which area is larger - pink or blue? =-- ## Solution by [[Pythagoras' Theorem]] and [[Area of a Circle]] +-- {.image} [[TwoRectanglesandaSquareIILabelled.jpeg:pic]] =-- Take the side length of the squares as one unit. Applying [[Pythagoras' theorem]] to triangle $O A B$ shows that the radius of the pink semi-circle is: $$ \sqrt{1^2 + 2^2} = \sqrt{5} $$ So from the [[area of a circle]], the area of the pink region is: $$ \frac{1}{2} \pi (\sqrt{5})^2 - 6 = \frac{5}{2} \pi - 6 $$ Applying [[Pythagoras' theorem]] to triangle $A C D$ shows that the radius of the blue quarter circle is: $$ \sqrt{1^2 + 3^2} = \sqrt{10} $$ So the area of the blue region is: $$ \frac{1}{4} \pi (\sqrt{10})^2 - 6 = \frac{5}{2} \pi - 6 $$ Therefore, the pink and blue regions have the same area.