FreeCAD: This is NOT a square!

Added: 2026-05-19

[00:00:06]I saw a geometry joke posted on two reddits the other day.

[00:00:11]Looking at this diagram, I have to say no, it's not a square.

[00:00:16]It's funny and perhaps it's even a bit of a geometric pun.

[00:00:20]At the grade school level, it's clearly not a square.

[00:00:24]I thought about it a little bit to see if in some weird geometry it could possibly be considered a square.

[00:00:31]I don't think so.

[00:00:33]Some of the angles are inside and some of them are outside and no amount of weird space warping is going to change that.

[00:00:41]But the question wasn't actually do you agree that this is a square?

[00:00:45]The question was can you construct it at all?

[00:00:48]Sure, you can draw a diagram and declare all of the sides are equal and the angles are 90°.

[00:00:55]But can you actually construct it?

[00:00:57]I suspected that you could, but just for the fun of it, I decided to let the free CAD sketcher have a go.

[00:01:04]Start with a sketch on the XY plane.

[00:01:07]Put the initial arc on the origin.

[00:01:14]Draw the two straight sides of the square and use an arc and three points to close it at the top.

[00:01:24]Set the two lines to be at right angles to the arc.

[00:01:31]It looks like somehow my intended coincident constraint ended up point on line, so I'll fix that now.

[00:01:39]Now constrain the larger arc at right angles to the two lines.

[00:01:45]We need to constrain all of the sides to be of equal length.

[00:01:50]But the equality constraint isn't going to do it.

[00:01:53]We need finer control over what should be equal.

[00:01:58]Fortunately, that can be managed.

[00:02:01]Select the dimension tool and the arc at the origin.

[00:02:06]It begins trying to set a radius dimension.

[00:02:09]But much like the polyline tool, I can press M to go to different modes.

[00:02:15]Once gives me the diameter.

[00:02:18]A second press goes to the angle of the arc.

[00:02:22]And finally, a third press gives me the length of the curve. That's the one.

[00:02:27]The symbolic representation of it is a bit confusing, but the key point is to notice the arrows at the end pointing inward indicating that it is the length of the curve and not the length of the missing imaginary segment of the circle.

[00:02:44]Let's go with 75 mm.

[00:02:48]But in order to set equality, the constraint needs to have a name so it can be referred to elsewhere.

[00:02:55]I'm going to call it side len.

[00:02:58]Now set the length of the side using the dimension tool.

[00:03:02]Make it a formula constraints.side len.

[00:03:07]It's not quite the same as an equality constraint since it can only be driven in one direction from the arc to the line, but it's close enough for this purpose.

[00:03:20]Note that when I set the length of the other side, the constraint just goes away.

[00:03:25]Sometimes the solver thinks harder than I do.

[00:03:28]It has clearly realized that because of the right angle relationships with the two arcs, the sides are necessarily of equal length, so the constraint is redundant.

[00:03:40]Finally, set a constraint on the length of the larger arc.

[00:03:45]Again, set the formula to And there we are.

[00:03:50]Proof that the diagram shape can actually be constructed.

[00:03:55]There is still one degree of freedom, the rotation.

[00:03:59]I'm not really concerned about that, so I'll just leave it.

[00:04:03]The question also asked if the two lines would really intersect at the center of the arc.

[00:04:09]I don't need the solver for that one since the lines are normal to the arc, they will necessarily pass through the center.

[00:04:18]Thank you for watching. If you like this video or found it useful, please like, subscribe, and share. If there's anything you'd like to see covered here, please let me know in the comments below.