The three angles of the triangle add to $180^{\circ}$, so the combined area of the three sectors of the circles that are inside the triangle add up to half a circle with area:

$\frac{1}{2} \times \pi \times \times 2^2 = \frac {4\pi}{2}= 2\pi$.

So the grey area is $(80- 2\pi) cm^2$.

*This problem is taken from the UKMT Mathematical Challenges.*