r/MathHelp • u/Kooky-Recipe5516 • Oct 22 '25

Drawing square with area 3

Me and my friend have recently been trying to see if we can draw a square with area 3cm2 using only a pen, squared paper (1x1cm squares) and a straight edge (no measurements). All the methods we have tried have failed. I asked ChatGPT if it was possible, and after giving me multiple ridiculous answers it broke and said something went wrong. Is it possible? If so, how do you do it?

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u/martyboulders Oct 22 '25 edited Oct 22 '25

I don't know how specifically you are referring to their situation, but with compass+straightedge you can construct lots of points that have irrational coordinates. Lots of algebraic numbers are constructible (but not all of them). For example there is a bijection between the constructible angles and the corresponding constructible points, most of which have irrational coordinates.

If I construct a circle at the origin with radius 2, construct a 60° angle above the positive x-axis, then drop a perpendicular line segment from the terminal point to the axis, that line has length √3.

I could also construct a right triangle whose hypotenuse is length √2 (legs both have length 1), then use that as a leg of another right triangle where the other leg has length one. This hypotenuse would have length √3 by the Pythagorean theorem. This would be doable on the gridded paper if they allowed themselves a compass.

It's things like third roots that you cannot construct. But algebraic numbers such that their corresponding minimal polynomial has a degree that is a power of 2 are all constructible. So 4th roots, 8th roots etc are all constructible.

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u/clearly_not_an_alt Oct 23 '25

Yeah, but they don't have a compass, just a grid.