# Solution to the Triangle and Circle Inside Square Puzzle +-- {.image} [[TriangleandCircleInsideSquare.png:pic]] > The equilateral triangle in the corner of this square is tangent to the circle where they touch. What's the diameter. =-- ## Solution by [[Angles in an Equilateral Triangle]], [[Angle Between a Radius and Tangent]], [[Angles in a Triangle]], and [[Isosceles Triangles]] +-- {.image} [[TriangleandCircleInsideSquareLabelled.png:pic]] =-- 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 $2 \times 15 = 30$.