Notes
triangle and circle inside square solution

Solution to the Triangle and Circle Inside Square Puzzle

Solution by Angles in an Equilateral Triangle, Angle Between a Radius and Tangent, Angles in a Triangle, and Isosceles Triangles

With the points labelled as above, first note that point $F$ lies on a diagonal of the square since the circle touches two sides. Therefore, angle $B \hat{A} F$ is $45^\circ$. Since the interior angle in an equilateral triangle is $60^\circ$, angle $B \hat{A} C$ is $30^\circ$ and so angle $C \hat{A} F$ is $15^\circ$.

As the angle between a radius and tangent is $90^\circ$, angle $F \hat{C} A$ is $90^\circ + 60^\circ = 150^\circ$. This leaves $180^\circ - 150^\circ - 15^\circ = 15^\circ$ for angle $C \hat{F} A$ as the angles in a triangle add up to $180^\circ$. So triangle $F C A$ is isosceles and hence $C F$ and $A C$ have the same length. The diameter of the circle is therefore $2 \times 15 = 30$.