Area at centre of Venn circles

Question

How much information do we need to calculate the area of the centre of $3$ Venn circles?

I would guess we need to know the lengths of the sides of the triangle formed by the circle centres and the circle radii.

Answer 1

You mention six parameters for the given area: the three sides of the triangle, and the three radii of the circles. There are many ways to set parameters that will defined the area, but they will have six degrees of freedom, as in the parameters you list.

We can see more easily from my diagram that six parameters are needed. Instead of the radii, I note the lengths of the line segments from the midpoints of the triangle's sides to the corresponding circles. You can think of each segment as an indicator of the curvature of the circle: if the triangle side stays the same and the circle' curvature increases, so does the length of the segment.

We consider six parameters here: the triangle's sides and the segments lengths. We see that the area of the intersection of the circles is divided into four regions: the triangle, and the circle segments on the triangle' sides.

If we increase the side of one triangle and leave the other five parameters unchanged, the area of the overlap increases. If we increase the length of one of the line segments and leave the other five parameters unchanged, the area of the overlap again increases. We therefore see that just five of those six parameters are insufficient to determine the overlap area, and the six parameters are independent of each other.

We conclude that the area of overlap indeed has six dimensions of freedom.