The triangle made up of the stand up arcs has area $\frac{1}{2}\times r \times \frac{1}{2}\pi r$ which is $\frac{1}{4} \pi r^2$. This is equal to the area of a quarter of the unit disc so the unit disc has area $\pi r^2$.
Suppose you straightened complete circuar arcs.Then the tops would lie on the line $y = 2 \pi x$ and the area of the triangle made from these stand up arcs would be $\frac{1}{2}\times r \times2 \pi r$ which is $\pi r^2$. This gives the area of the whole unit disc.