# Solution to the Semi-Circle in a Right-Angled Triangle Puzzle +-- {.image} [[SemiCircleinaRightAngledTriangle.png:pic]] > A semi-circle sits in a right-angled triangle. What's the shaded area? =-- ## Solution by [[Angle Between a Radius and Tangent]] +-- {.image} [[SemiCircleinaRightAngledTriangleLabelled.png:pic]] =-- With the points labelled as in the diagram above, the point labelled $O$ is the centre of the circle. Angles $O \hat{A} C$ and $C \hat{B} O$ are both [[right-angles]] as they are the [[angle between a radius and tangent]]. Angle $A \hat{C} B$ is given as a right-angle. Therefore, [[quadrilateral]] $A O B C$ is a [[rectangle]]. The lengths of $O A$ and $O B$ are equal as they are radii of the semi-circle, so in fact $O A B C$ is a [[square]]. As the diameter of the semi-circle is $8$, the side length of the square is $4$ and its area is $16$. The triangle $A B C$ is half of that square and so has area $8$.