Cutting a circular table

This Riddler Classic puzzle is about cutting circles out of rectangles!

You’re on a DIY kick and want to build a circular dining table which can be split in half so leaves can be added when entertaining guests. As luck would have it, on your last trip to the lumber yard, you came across the most pristine piece of exotic wood that would be perfect for the circular table top. Trouble is, the piece is rectangular. You are happy to have the leaves fashioned from one of the slightly-less-than-perfect pieces underneath it, but there’s still the issue of the main circle. You devise a plan: cut two congruent semicircles from the perfect 4-by-8-foot piece and reassemble them to form the circular top of your table. What is the radius of the largest possible circular table you can make?

Here is my solution to the case of a general rectangular table. The result may surprise you!
[Show Solution]

5 thoughts on “Cutting a circular table”

  1. For a moment I was hoping the crossover point was 4/phi, but it looks like it’s really (7 – 17^0.5 + (62 – 14*17^0.5)^0.5)/2

  2. What is meant by two congruent semicircles? Look at the flag of the Republic of Korea. There you will see a circle divided into two congruent pieces. Each piece is bounded by a semicircle and a wavy line joining the two ends of the semicircle. Are these congruent pieces congruent semicircles? I argue that they are, because the word semicircle does not define an area, but only an arc. Congruent pieces dividing a circular object into N pieces must each include a continuous circular arc that divides the circular perimeter in N equal parts. They must also each include, on an edge, the center of the circular arc. But there is no reason to suppose that the pieces have to include straight radial lines joining the center to the ends of the arc.

    In the context of this Riddler, we can find a solution that gives a bigger circular table top than the one described in the posted solution. Only two congruent pieces are involved, and they can be made by one zigzag cut in the rectangle, a displacement that converts the shape into a square with two opposite corners snipped off, and then a circular cut with diameter 2^1.5 feet. This is larger that the given solution.

    1. You’re right — if you allow for congruent shapes that don’t necessarily include the radial lines, then you can fit a larger circular area inside the rectangle. I took “semicircle” to mean half of a circle as formed by a straight cut through the center, but your interpretation could lead to some very interesting solutions as well!

    1. It’s definitely a constrained optimization problem, e.g. you can parameterize the positions of the segments and their orientations and sizes, and ask yourself what choice of parameters maximizes the area. However, the problem is made very challenging because we must include constraints that prevent the segments from overlapping the boundary or each other. This renders the problem non-convex and probably hopeless to solve in general.

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