# Solution to the [[Two Perpendicular Equilateral Triangles]] Puzzle +-- {.image} [[TwoPerpendicularEquilateralTriangles.png:pic]] > The medians of these two equilateral triangles are perpendicular. What’s the area of the larger circle? =-- _Note:_ for this to be a well-defined problem, the equilateral triangles are assumed to be the same size. ## Solution by [[Similar Triangles]] and Lengths in an [[Equilateral Triangle]] +-- {.image} [[TwoPerpendicularEquilateralTrianglesAnnotated.png:pic]] =-- With the points labelled as above, triangles $A B C$ and $A D E$ are [[similar]], and the scaling also takes one circle to the other. The scale factor is found by comparing the lengths of $A B$ and $A C$. That of $A B$ is half the side length of one of the [[equilateral triangles]] while that of $A C$ is $\frac{\sqrt{3}}{2}$ of the side length. So the ratio $A B : A C$ is $1 : \sqrt{3}$. The [[area scale factor]] is therefore $3$ and so the area of the larger circle is $3$.