Notes
circle in triangle in square solution

Solution to the Circle in Triangle in Square Puzzle

Solution by Properties of Equilateral Triangles

From lengths in an equilateral triangle, the length of $O B$ is one third of $C B$, so $C B$ has length $3$. Then the length of $C B$ is $\sqrt{3}$ times the length of $B D$, so $B D$ has length $\sqrt{3}$. Triangle $A B D$ is isosceles and right-angled, so $A B$ has the same length as $B D$. The length of the diagonal of the square is then $\sqrt{3} + 3$. The area of the square is $\frac{1}{2}(\sqrt{3} + 3)^2 = 6 + 3 \sqrt{3}$.

Since the area of an equilateral triangle is $\frac{\sqrt{3}}{4}$ times the square of the length of one of its sides, the triangle has area $\frac{\sqrt{3}}{4} \times (2 \sqrt{3})^2 = 3\sqrt{3}$. The shaded area is therefore $6$.