# Solution to the Circles in Overlapping Triangles Puzzle +-- {.image} [[CirclesinOverlappingTriangles.png:pic]] > Two overlapping equilateral triangles each have an area of $100$. The three circles are the same size. What's the total area of the design? =-- ## Solution by [[Lengths in an Equilateral Triangle]] and [[Area Scale Factor]] +-- {.image} [[CirclesinOverlappingTrianglesLabelled.png:pic]] =-- In the above diagram, the points labelled $C$ and $E$ are the centres of their circles, while $G$ is such that angle $C \hat{G} A$ is a [[right-angle]]. The line that extends $A C$ cuts the red triangle in half, so angle $G \hat{A} C$ is $30^\circ$ which shows that triangle $C G A$ is half an [[equilateral triangle]]. This means that the length of $A C$ is twice that of $C G$, so the length of $A B$ is the same as the radius of the circles. The length of $A E$ is therefore $4$ of these lengths, and $A F$ is $5$ of them. Cutting the red equilateral triangle at $E$ gives a triangle $\frac{4}{5}$ths of the height of the red one, which therefore has area $\frac{16}{25}$ths of the red as the [[area scale factor]] is the square of the [[length scale factor]]. Since the red triangle has area $100$, this shorter triangle has area $64$. Triangle $J I H$ is also [[equilateral]] and its height (of $J$ above $I H$) is one radius of the blue circles, which is one fifth of the red triangle. Its area is therefore one twenty-fifth of the area of the red triangle, which is $4$. The total area of the design is therefore: $$ 64 + 64 + 4 + 4 + 4 + 4 = 144 $$