Mrs. Miniver's problem

of given radii in such a way that the lens formed by intersecting their two interiors has equal area to the symmetric difference of

[1] It was named for an analogy between geometry and social dynamics enunciated by fictional character Mrs. Miniver, who "saw every relationship as a pair of intersecting circles".

The problem derives from "A Country House Visit", one of Jan Struther's newspaper articles appearing in the Times of London between 1937 and 1939 featuring her character Mrs. Miniver.

Beyond a certain point the law of diminishing returns sets in, and there are not enough private resources left on either side to enrich the life that is shared.

Probably perfection is reached when the area of the two outer crescents, added together, is exactly equal to that of the leaf-shaped piece in the middle.

[2]Louis A. Graham and Clifton Fadiman formalized the mathematics of the problem and popularized it among recreational mathematicians.

If the ratio of radii falls beyond these limiting cases, the circles cannot satisfy the problem's area constraint.

The rhombus formed by the two circle centers and the two crossing points, with side lengths equal to the radius, has an angle

Three instances of Mrs. Miniver's problem. In each case the inner yellow area equals the total area of the surrounding blue regions. The left case shows two circles of equal areas, the right case shows one circle with twice the area of the other, and the middle case is intermediate between these two.