# Stacked Shapes +-- {.image} [[StackedShapes.png:pic]] > The triangles are equilateral, and the rectangles are twice as long as they are wide. If the area of the semicircle is $4$, what's the area of the pink circle? =-- ## Solution by [[Lengths in an Equilateral Triangle]] +-- {.image} [[StackedShapesLabelled.png:pic]] =-- As clarified in the twitter thread, shapes of the same colour are [[congruent]]. The side of the equilateral triangle has the same length as the short side of the rectangle, and so half the length of the long side. Let $a$ be this length. Then $B D$ also has length $a$. Angles $D \hat{B} A$ and $E \hat{D} B$ are both the [[exterior angles]] of an [[equilateral triangle]] so are both $120^\circ$. Angle $O \hat{D} C$ is half that angle, so triangle $O D C$ is half an equilateral triangle. From the relationships between [[lengths in an equilateral triangle]], $O C$ has length $\frac{\sqrt{3}}{2} a$. Triangle $D Q F$ is also half an equilateral triangle, since $D F$ has length $a$, $F Q$ has length $\frac{1}{\sqrt{3}} a$. The area of the semi-circle is then $\frac{1}{6} \pi a^2$ so $\pi a^2 = 24$. The area of the circle is therefore $\frac{3}{4} \pi a^2 = 18 \pi$.