square in circle?

[Kerry Montgomery]

Hi all, Was working on marking a disk into three equal pie sections, and was offered a suggestion that I could put a square (4 sided figure, not a carpenter's square) on the circle (maybe an inscribed square) and that, by rotating that square, it would make finding the thirds of the disk easier or more foolproof. This suggestion was made by a boatbuilder/woodworker, and I have to admit that I couldn't see how this would help me. Is there a method of using a square on a circle that does make dividing the circle easier? Thanks, Kerry

[Old Guy]

I'm don't know how that would work.

If you measure the radius, then start from a point on the circumference, and set off a chord (a straight line that touches the circumference at both ends, equal to the radius, and from the end of that chord set off another one, the end of the second chord will be on a point 1/3 of the circumference.

Old guy

[Kerry Montgomery]

If you measure the radius, then start from a point on the circumference, and set off a chord (a straight line that touches the circumference at both ends, equal to the radius, and from the end of that chord set off another one, the end of the second chord will be on a point 1/3 of the circumference.

Old guy

Old Guy, Yep, that's what I did all right. This fellow sounded like he used this square in a circle technique fairly often, so was wondering if it might be some secret bit of knowledge that I hadn't come across. Thanks, Kerry

[Rich]

It could be a gannin square? It has 9 points so picking 3 would be a snap?

Rich

[David G. Nagel]

Set your compass to the radius of the circle. Pick a point. Scribe a line with the compass from one edge of the circle to the center and to toe other edge. Move the compass to one of the intersections just scribed and repeat the action. Continue until you return to the first point. Pick every other intersection and scribe a line from the intersection to the center. You will have your three EXACT wedges. Draw your square.

Dave N

[Bill]

It would work since there is a hexagon, comprised of 6 equilateral triangles sharing a vertex in the center of the circle with the other vertices on the boundary of the circle. The length of the sides of every one of these triangles is the same as the radius of the circle. Draw a picture. Then note that if you chose every other vertex on the boundary of the circle, they will form an equilateral triangle.

Measured along the boudary, the distance between the 3 vertices of an inscribed equilateral triangle should be Pi times D , where Pi ~ 3.1415... and D is the diameter.

Both of these techniques assume you have a "perfect" disk/circle to work with. I would verify that the (shortest) distance between each of them is the same before I cut anything.

I hope that something I wrote is helpful! Bill

[David Nebenzahl]

On 5/18/2009 4:02 PM David G. Nagel spake thus:

Not *quite* exact; your method uses the fact that the relationship between a circle's circumference and diameter is pi, about 3.14159.

I know about this 'cuz I was just rereading the /Fine Woodworking/ book of "Proven Shop Tips". One of them is a table for dividing a circle into equal parts. For three equal parts, take the diameter of the circle and multiply it by 0.866. Set your dividers to the resulting size and "walk" it around the circle to evenly divide it. (There's a table in this tip that goes up to 100 parts.)

[Bill]

Once you draw a square in the circle, then you could draw both diagonals the find the center (of course, you probably already know where the center is by the time you've drawn a square!). Use your compass to pick up the measure of the radius, and starting anywhere, scribe 6 consecutive arcs along the boundary of the circle (they are the vertices of a hexagon as I described in my previous post). Choose every other one to obtain the vertices of a triangle.

Bill

>

[David Nebenzahl]

On 5/18/2009 6:15 PM Bill spake thus:

Not quite. As you yourself point out, pi != 3. Close, but no cigar.

[Keith Nuttle]

Your Geometry teachers are rolling over in the grave if they ever become aware of this discussion. The most accurate way to divide a circle three parts is to use the radius on the circumference technique.

or as stated above >>Set your compass to the radius of the circle. Pick a point. Scribe a >> line with the compass from one edge of the circle to the center and to >> to other edge. Move the compass to one of the intersections just >> scribed and repeat the action. Continue until you return to the first >> point. Pick every other intersection and scribe a line from the >> intersection to the center. You will have your three EXACT wedges.

[Doug Miller]

He's absolutely right. Read it again -- draw it if you need to.

[Doug Miller]

No, it *is* exact, and in no way relies on the fact that pi is close to 3.

Rather, it relies on the facts that a circle can be circumscribed about any regular polygon, and that a regular hexagon can be decomposed into six equilateral triangles. Draw it yourself if you don't believe me.

Ummmmm..... no.

[Kerry Montgomery]

Hi David, The radius method is exact, as the radius is being used as chords across the circle, not following along the circumference. Thanks, Kerry

[Lew Hodgett]

Dave's Hypothesis states:

Dave's Hypothesis is incorrect.

Basic plane geometry proofs do not allow calculated values to be used in the proof.

For the subject under discusion, the actual value of circumference is an innocent bystander, not a player

Lew .

[Lew Hodgett]

Neither do I.

Lew

[Dan Coby]

Ummmm....yes.

If you mark off chords whose length is 0.866 (actually sqrt(3)/2) times the diameter, then you will get an equilateral triangle.

[John G.]

The the sides of the six triangles are straight lines and hence a bit shorter than the 6th part of the circumference so are closer to a good fit than Pi would imply. John G.

[Doug Miller]

Yes, of course the sides of the triangles are shorter than the arcs -- but surely it's clear that the vertices of the hexagon divide the circle into six exactly equal parts, no?

[Doug Miller]

You're right -- my mistake. Had a brain fart doing the math...

[basilisk]

I've uploaded a simple excel spreadsheet that will calculate the chords for any divisions of a circle "bolt circle.xls, it is here:

open directly in excel if you have it installed. I haven't tried it in open office.

basilisk