# Solution to the Triangle in Semi-Circle with Square Puzzle +-- {.image} [[TriangleinSemiCirclewithSquare.png:pic]] > The equilateral triangle sits in the centre of the semicircle. What's the area of the square? =-- ## Solution by [[Lengths in an Equilateral Triangle]] and [[Pythagoras' Theorem]] +-- {.image} [[TriangleinSemiCirclewithSquareLabelled.png:pic]] =-- With the points labelled as above, the radius of the semi-circle is $3$ and this is the height of the [[equilateral triangle]]. From [[lengths in an equilateral triangle]], the length of $O A$ is $\sqrt{3}$. Applying [[Pythagoras' theorem]] to triangle $O A B$ shows that the square of the side length of the square is $3^2 - 3 = 6$ and so the area of the square is $6$.