Divide a circle into three equal parts [closed]

You are at a restaurant. There are a total of three people(including you). Now you order a pizza and the trouble is that all three(including you) want equal parts. You have only a knife.

How do you cut the pizza in three equal parts?

• Please state the rules more clearly. Like: "You are only allowed to make one cut" "You are not allowed to do any measurements" "You are only allowed to make straight cuts during the complete slice youre cutting" Something like that. The sentence "You only have one knife." doesn't really say anything. Nov 18, 2015 at 14:09
• Do whatever, just use a knife. :)But remember, you are at a restaurant, and I don't think you take any measurements there. my friend gave this puzzle to me.
– 4-K
Nov 18, 2015 at 14:09
• Please say I'm allowed to use a piece of dental floss, or that i'm allowed to assume perfect eyesight and perpendicular cutting skills. :) Nov 18, 2015 at 14:45
• To clarify, does the pizza need to be cut into exactly 3 equal pieces, or just equally divided between 3 people (i.e. 6 slices, with 2 to each person)? Nov 18, 2015 at 15:01
• One person takes the knife and divides the pizza into three parts. The other two doesn't dare argue with the knife wielder. Aug 17, 2016 at 9:07

Solution

Take your pizza and fold it like so to get the proper lines in the crust

As you can see it is now easy to equally divide your pizza into three parts.
Only having a knife is no problem now!

EDIT for even edges

There are many origami techniques to get the pizza to be divided evenly like in the steps shown below.

Once the edges line up properly and evenly, while folding the half circle into a "Z" type shape, that's when you can make a good crease into the crust and can then proceed with the knife!

• How do you get the angle? Nov 18, 2015 at 14:37
• He doesn't. You can fairly easily construct them, with at little as a piece of dental floss though. If you don't even get that, I'm not sure how you can get them at all with any sort of accuracy. Nov 18, 2015 at 14:46
• @kaine Got you covered on the angles! Nov 18, 2015 at 16:28
• But ... won't your toppings all get smushed together and ruin the whole point of getting a pizza instead of a panzerotti? Nov 18, 2015 at 21:10
• @IanMacDonald He said "Do whatever, just use a knife." I just want him to share that pizza as equally as possible! Plus this is about the only thing I could think of using absolutely no tools or anything or assuming anything. Nov 18, 2015 at 21:15

At the risk this is not the intended solution, but here goes.

We are assuming:

• you know the exact center point of the pizza
• you have a piece of clean string, equal to at least the pizza's radius. (dental floss?)

Cut a straight line from the center of the circle to the outer edge. where the line would extend to the other outer edge, don't cut the entire line, but make a marking cut on the outer edge. Now take your piece of string, and 'measure' the radius of pizza (along the cut you first made). Using said lenght, use that as a radius, from the 'marking cut' you made on the opposite side. You can use that to find two intersecting points with a circle radius r from the 'marking cut' point with the circles edge. Mark those two points with a small cut. Now just make straight cuts from both of those new 'marking cuts' to the center.

Why does this work?

Cosine(60) = 1/2. Basically, the perpendicular bisector of a line cuts it in half and is easy to construct. The second set of marking cuts have a cosine of 1/2 (half the radius compared to radius). 2x that angle makes a 120 degree angle between the parts, which is what you need to form a perfect three way split on a 360 degree circle.

Video of a similar method:

(but using measuring to spot the half, which I conveniently replaced with the 'use floss to create the perpendicular bisector' thingy): Youtube video

EDIT: This would also work if you think you have proficient eyesight to spot the middle of a line perfectly, and can but perfectly perpendicular without any assistance.

Different method then other answer, but works equally well using a piece of string / dental floss / whatever.

We are assuming:

• you know the exact center point of the pizza
• you have a piece of clean string, equal to at least the pizza's radius. (dental floss?)

Method:

From the center of the circle, use the string to 'copy' the radius of the pizza. Pick a random point on the other edge, make a marking cut. Use the string to find the intersections of a circle with radius r with the edge of the pizza (two such points), mark them with cuts. From one of the cuts, repeat the method, you'll find two points (the one you came from, and one new). Repeat two more times to find a total of 6 cuts. Then, cut in straight lines from center to cuts on the outside, each time skipping one of the cuts. You'll end up with 3 120 degree parts.

Graphic version:

Of course you don't cut the triangle, but lines from center to the 'triangle points' on the outer edge.