The Table Problem

Algebra Level 3

For a project, Tom is making a tablecloth for a circular table 3 feet in diameter. The finished tablecloth needs to hang down 5 inches over the edge of the table all the way around. To finish the edge of the tablecloth, Tom will fold under and sew down 1 inch of the material all around the edge. Tom is going to use a single piece of rectangular fabric that is 60 inches wide.

*Picture only for viewing purposes only, don't use for scaling *

What is the shortest length of fabric, in inches, Tom could use to make the tablecloth without putting any separate pieces of fabric together?

30 24 48 15 42

Rui Chang Lu by Rui Chang Lu , 15, Oman

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2 solutions

Rui Chang Lu
Oct 18, 2019

Before doing anything else, make sure you convert all your measurements into the same scale. Because we are working mainly with inches, convert the table with a 3-foot diameter into a table with a 3 * 12 = ( 36 ) inch diameter .

Now, we know that the tablecloth must hang an additional 5+1 inches on every side, so our full length of the tablecloth, in any straight line, will be:

1 + 5 + 36 + 5 + 1 = 48 inches .

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Richard Desper
Oct 17, 2019

From the circular top of the table cloth is 36 inches wide. Add in 6 inches for the hanging part on each side and 1 inch for each hem to get 48 inches.

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Does the geometry actually work out for this, though? It seems there would be issues with curvature, much like the problem of creating a flat map of a spherical planet.

One way around this is to cut the cloth and re-stitch it into the net of a cylindrical cap (one circular piece for the tabletop, one rectangular piece for the drop). But this isn't allowed in the problem.

The other alternative is accepting that - as in the picture - you'll get pleating in the tablecloth. There's no cutting in this version, but then is the drop 5 inches all the way round?

(I'm not even considering the hemming here - you couldn't actually fold a circular hem flat without cutting or pleating)

Chris Lewis - 1 year, 7 months ago

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Ok, I will post up a solution for this.

Rui Chang Lu - 1 year, 7 months ago

Certainly it's impossible to cut a tablecloth of this shape from a flat sheet isometrically. In a flat sheet, the radii of concentric circles increases as one moves further from the center. But with a table cloth, the radii only increase along the top disk. After that, the radii are constant: in this case we have many circles with 18" radius.

This fact is why sewing is required to make tablecloths. That is, if one wishes to have a constant drop on each side of a round table.

Richard Desper - 1 year, 7 months ago

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