How to calculate the intersection of the 2 different circle's area

Question

I have two different circles with their formulas, which intersect from different parts of themselves, how can I calculate the area of the intersection and the rest of the circles? Is there some way to do it without using calculus? And how to calculate that area, why? Here is the question, if you find a way, it doesn't matter the exact numbers:

Answer 1

1. Find the intersections (I will assume that you have two points)
2. From the coordinates of the intersections and the radii of the two circles you calculate the angle between the radii corresponding to the intersection points. So, for the purple circle, the angle between $(-\sqrt{12},0), (0,0), (\sqrt{12},0)$, and for the blue circle between $(-\sqrt{12},0), (0,2), (\sqrt{12},0)$. The sine of half of the angle is equal to the ratio between half the distance between the intersection points and the radius.
3. Calculate the area of the circular sectors defined by the line between the intersections https://en.wikipedia.org/wiki/Circular_segment#Arc_length_and_area