# Where’d I get this geometry puzzle?

April 23, 2011 pm30 1:20 pm

Embarrassed. I shared a puzzle, thought it was massively cool. And I want to give credit, but don’t recall where I grabbed it from. Any helpers?

Consider a square. Consider four circles, centered one at each vertex, with radius equal to the length of the side of the square. What is the area of the intersection of the circles (in terms of the radius)?

The original only gave the quarter circles within the square. I think it gave sidelength = radius = 1. There was a diagram included, maybe? And there was a word description of the region, something like a “bulging square”

You want to try solving this?

But more importantly, can you tell me who to credit?

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JD..

I found it in Index to mathematical problems, 1975-1979

edited by Stanley Rabinowitz, Mark Bowron.. contributed by Donald Chambers..

It’s on the bottom of page 125. I have also seen it described as a “traditional ” problem to do without calculus… Hope that helps

Of course the tougher extension might be…given eight unit spheres centered on the vertices of a unit cube… find the volume of intersection… And in case you wondered, I just made that one up, although I can’t believe I would be the first…

This is bothering me. I am having trouble visualizing the solid. Would it be a cub-oid, a cube with spherical cushions on each face?

It’s got to be small. With edge = radius = r, the length of the body diagonal is and the body diagonal of the cub-oid is something like , which isn’t much at all.

It might be Sue’s book review of Rediscovering Mathematics, though, I don’t see a reference to a bulging square.

That’s it! Thank you. I must have made up “bulging square”.

Yep, lovely problem.

With a unit square I got 1 – sqrt(3) + pi/3 but I haven’t really checked over my work. That’s a little more than 0.3 so it seems like it makes sense. Anyone want to verify?