The area of the largest triangle that can be inscribed in a semi-circle of radius r is:

The base of the largest triangle inscribed in a semi-circle is the diameter of the semi-circle, which is $2r$. The height of this triangle, when the vertex is at the midpoint of the semi-circle arc, is equal to the radius $r$. Therefore, the area is $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times (2r) \times r = r^2$.