Area of circle showing concavity

Question

How to find the area of circle showing the change of concavities of function $$\displaystyle f(x,y)=1-e^{-(x^2+y^2)/2}$$

Answer 1

The red circle shows where the curvature on the function changes sign. (Perhaps this is what the OP seeks... hard to know.)

Here's a cross section (in blue) and the second derivative (in tan), which changes sign at radius = 1.