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You are given a pen and a large piece of paper. Without using any additional tools (eg., ruler, compass, string etc) how can you draw a perfect circle?

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19 Answers 19

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My solution:

Make a compass:

Grasp the pen in your fist so that your thumb is uppermost and the pen points down, like holding a dagger.

Using the same arm, place your elbow's funny-bone at the centre of the large sheet of paper.

Rest the tip of the pen on the paper.

Keeping your arm still, drag the paper round with your other hand so that it rotates under your funny-bone.

The pen will draw a line as the circumference of a circle. Its radius is the length of your forearm.

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    $\begingroup$ Yes that's it! This is the answer I had in mind. Well done. $\endgroup$ – Dmitry Kamenetsky Oct 10 at 10:38
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    $\begingroup$ For smaller circles, I have done the same using my pinky as an "anchor" at the centre of the circle, holding the pen between thumb, index finger and (optionally) middle finger. $\endgroup$ – mishan Oct 10 at 12:36
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    $\begingroup$ @WeatherVane That's how I do it. I anchor my pinky and rotate the paper around it while holding the pen still using an acceptable number of remaining fingers. $\endgroup$ – mishan Oct 10 at 13:09
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    $\begingroup$ This would never work. At best you get a wobbly piece of crap. Send photos to prove me wrong. $\endgroup$ – Frank Oct 10 at 21:38
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    $\begingroup$ It needs somewhere to place the rest of your body that will allow the paper to turn...the parts of your body that touch the floor must be at least a forearm's length from your elbow. I have just laid with my hip and leg on the floor and there is so much pressure on my elbow which is supporting my trunk, other arm, head and shoulders that I don't think I could rotate the paper without tearing it. And as I only have a newspaper of large enough size, that's certain ;) I think the best answer was the one from Paul Panzer. $\endgroup$ – Weather Vane Oct 11 at 13:04
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Oh, I'm obviously thinking too far outside the box, but this is a cute solution:

Draw a straight line across the page. When the page is rolled up, the straight line becomes a circle:

solution

Objection #1: The circle is imperfect because it is hard to draw a straight line.
Response: This is true to some degree of all the solutions, but we can mitigate by using the bottom edge of the paper as a rule to keep the line straight.

Objection #2: The circle is imperfect because it is hard to roll the paper into a perfect circle.
Response: Keep rolling it until the circle becomes more perfect.

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    $\begingroup$ Nice creative solution. I think the first objection can be mitigated by folding the paper in half and drawing the line at the crease - it should be quite straight. $\endgroup$ – Dmitry Kamenetsky Oct 10 at 12:14
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    $\begingroup$ You could tear half the paper and then use it to make a cylindrical template to draw around $\endgroup$ – Frank Oct 11 at 7:10
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If the piece of paper is rectangular and not too rigid:

You can draw two points which will be one diameter and then use a corner of the sheet and Thales's Theorem to draw as many points on the circle as you like. enter image description here

A bit more detail for the non-geometers: Thales's theorem says that if you connect a point on a circle with the two ends of a diameter you will get a right angle and, conversely, if a point forms a right angle when connected with the ends of a diameter then the point lies on that circle. Therefore after choosing two points which shall be our diameter we use the corner of the sheet as a right angle and gently bend the sheet so that the two sides enclosing the corner align with our two points. We can do this in many ways and each will give us another point on the circle.

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    $\begingroup$ Very nice! Is there an easier way? $\endgroup$ – Dmitry Kamenetsky Oct 10 at 6:07
  • $\begingroup$ The easier way is to draw it on the wall, not on the paper itself. $\endgroup$ – Florian F Oct 10 at 20:37
  • $\begingroup$ I'd just rip off a corner of the paper to make my own set square. Pretty smart idea though. $\endgroup$ – MaxD Oct 11 at 1:00
  • $\begingroup$ Wouldn't this only draw half of the circle - as you will only get points to the one side of the diameter? To get the other half of the circle you would need to bend and align the other corner (top-left in your diagram). $\endgroup$ – Dmitry Kamenetsky Oct 11 at 13:01
  • $\begingroup$ @DmitryKamenetsky yes, that's correct. $\endgroup$ – Paul Panzer Oct 11 at 14:38
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just put a dot, a circle of radius zero

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  • $\begingroup$ very clever! haha $\endgroup$ – Dmitry Kamenetsky Oct 10 at 3:44
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    $\begingroup$ unfortunately it's also a perfect square. $\endgroup$ – Jasen Oct 10 at 4:28
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    $\begingroup$ Unfortunately? It's two for the price of one :-) $\endgroup$ – Kramii Oct 10 at 10:07
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    $\begingroup$ I will be very impressed if you can make it exactly zero using a real pen. $\endgroup$ – Tanner Swett Oct 10 at 16:04
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    $\begingroup$ doesn't work - the shape of the dot can be far from circle $\endgroup$ – TMS Oct 10 at 16:59
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If the paper is large enough, you can use yourself as a compass. For example, you could make a mark in the centre of where you want the circle, and anchor a finger on it. With the pen in the other hand, and at full stretch, rotate about the anchored finger, leaving a circular trail as you go. You mighy want to experiment with other body parts to see which gives the best results. I reckon my nose would make a good anchor...

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  • $\begingroup$ I remember learning this trick a long time ago, I think in school. You can easily do this on a piece of A4 paper using your little finger knuckle comfortably. $\endgroup$ – Ambo100 Oct 10 at 14:47
  • $\begingroup$ Very versatile! $\endgroup$ – ention everyone Oct 11 at 20:49
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For an easy(?) way: 4 times this will do

enter image description here

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  • $\begingroup$ How do you ensure that the movement follows the boundary of the circle precisely? $\endgroup$ – Dmitry Kamenetsky Oct 10 at 10:37
  • $\begingroup$ 1 keep the moving line flat; 2 the horizontal fold will give some resistance to vertical movement of the movement center, (in my experience enough). $\endgroup$ – Retudin Oct 10 at 10:48
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This answer is very different from the others and I'm not sure if it's too obscure, but technically it fits the rules I think.

Draw 5 stick figures. One holding a microphone, two holding a guitar, one holding a bass guitar and one sitting on a drumset. Underneath those, write the names Maynard, Billy, James, Matt and Jeff. You have now drawn A Perfect Circle.

It's debatable how many people would recognize this band. However if it is recognized, it arguably fits the criterion of being "a perfect circle" better than any attempt to draw an actual circle since those can never be absolutely perfect.

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  • $\begingroup$ For this to be allowed, "A Perfect Circle" would've to be spelled with capital letters, though :) $\endgroup$ – MaxD Oct 11 at 1:01
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    $\begingroup$ ah right didn't think about that. Although their logo spells the name in all lowercase, so there's some inconsistency there $\endgroup$ – dave Oct 11 at 1:04
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If it's the right type of pen, I would just

place the tip of the pen on the paper and hold it there for several seconds. As the ink wicks into the paper, it will spread out in a perfect circle.

Demonstration:

enter image description here

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    $\begingroup$ Are you sure? Doesn't paper have some fibrous stuff in it that would make the ink spread unevenly? $\endgroup$ – Paul Panzer Oct 10 at 17:13
  • $\begingroup$ The same could be said for all of these answers. Even if you use a compass, the paper has fibrous stuff in it that will make the ink flow out of the pen unevenly and you'll end up with an imperfect circle. Whether my answer works or not depends on just how perfect you want to get. $\endgroup$ – Tanner Swett Oct 11 at 13:12
  • $\begingroup$ Never seen a compass that uses ink ;-) $\endgroup$ – Paul Panzer Oct 11 at 14:42
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You can

Write "A Perfect Circle" on the paper with the pen.

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    $\begingroup$ Or you can draw a perfect circle. It doesn't look perfect? Well, the drawing is not perfect. But it represents a perfect circle. It gives an idea of what is meant. $\endgroup$ – Florian F Oct 10 at 20:45
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Just as an "off the top" answer to this scenario that should be fairly close to perfect:

(I want to take a moment as perfection is often something we don't realize how varied peoples accepted notions for this can be. One person may believe 100% is perfect, another may need the 100% to be done while 100% of other factors are in place a well, etc. That said, this response has taken some basic liberties such as leaving out the variance in paper thickness, airflow of the ink drying and the ink's viscosity due to a variety of factors from chemical elements of the ink itself to the temperature in the pen/around the pen/at the point of the paper, etc. The list goes on and on but I believe the point has been made.)

Simply bend the pen in a manner to where you can mark on the paper with the writing end and pin (hold in place or create a pivot if you will, not the utensil itself) the other end somewhere on the paper which would allow the pin to spin a full circle without leaving the paper surface. Now rotate the pen 360 degrees (or spin the paper but this is getting long winded) and you should have a 'perfect' circle.

p.s. A Perfect Circle is a vastly under rated band. Just saying :P.

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easy!

Take the cap off the pen, put it down vertically on the paper (you may need to break off the little clip on it), and trace around it.

If your pen doesn't have a cap, simply take it apart, remove the ink tube and tip, and use that to trace around the pen's body.

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  • $\begingroup$ very original! Are the caps perfectly round? $\endgroup$ – Dmitry Kamenetsky Oct 11 at 2:08
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    $\begingroup$ @DmitryKamenetsky, with most pens I've used at least, yes! $\endgroup$ – Patrick L Oct 11 at 2:09
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A bit late to the game, but here's an answer that's "way out there":

Tear off a small piece of the paper, poke a tiny hole in the small piece of paper with the tip of the pen, and hold it up to the sun. Then trace the image of the sun's edge that comes through the "pinhole" lens of the tiny hole that projects onto the larger remaining piece of paper. You'd need to make sure that the large piece paper is perfectly perpendicular to the sun's rays, but per https://science.nasa.gov/science-news/science-at-nasa/2008/02oct_oblatesun, the sun is just about the roundest item in the universe.

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  1. Pick your writing implement carefully. It needs to be a simple round pencil, not one of those yellow octagonal pencils with a rubber on the end, nor one of those weird triangular ones that mysteriously appear in your pencil case, nor even one of those fancy mechanical pencils that your grandfather gives you. No, it needs to be a nice, uncomplicated round pencil.

  2. Press hard on the piece of paper to break off a little of the lead. This will leave a mark, but unless you're exceptionally lucky, it won't be a circle. That's fine because nobody said you can't leave non-circular marks on the paper.

  3. Now, use the blunt end of the pencil to turn the broken-off piece into powder. Take your time over this, not because you need to for the result, but because there's great pleasure to be taken in grinding things up. It may be a bit messy, but it will all be worth it in a moment.

  4. Make sure the blunt end of the pencil is completely coated in powder, and then press it against the paper. When you remove it, you'll have a perfect circle.

  5. Fame and fortune will inevitably follow as reward for your efforts. Well done, you!

I know, I know: the OP said I had to use a pen... well, I reckon something similar would work with a pen, too. Perhaps squeeze out a little ink, and take the pen apart and use half of it to print little circles?

I'll bet my granfather will be impressed with my ingenious solution. He might even reward me...

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    $\begingroup$ You got my vote for the comedy element :) $\endgroup$ – Dmitry Kamenetsky Oct 10 at 10:36
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Carefully rip off a 1 inch wide strip of the paper. Jam the tip of the pencil through near one end of the strip and through the sheet of paper near its center. Break off the pencil, resharpen, and then poke it through the strip at desired radius. Use the pinned strip to maintain constant radius as you pivot around.

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  • $\begingroup$ Oups thought pencil was valid tool after reading other answers . Never mind. $\endgroup$ – Dan Oct 10 at 21:09
  • $\begingroup$ If the pen has one of those plastic ink tubes inside could still break off a piece and use that as pivot anchor. $\endgroup$ – Dan Oct 10 at 21:12
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Eleven answers already! A record perhaps?

Here is my off base answer. I tried. Looked like a circle

I took a ball point pen and a thick paper. Just put the pen perpandicular to the paper and pressed hard to puncture the paper. The pen point was circular so it created a nice circular hole!

With a pencil it worked even better because the pencil end is conical in shape!

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Here's my solution:

Use a hair (you can get one on your head). Tie one end of the hair to around the point of the pen, and you got yourself a hassle free, easy to use compass!

Demonstration:

enter image description here

Extra info:

To use this compass, use your finger to press the hair at a certain distance from the end tied to the pen (distance depends on how big you want your circle to be), pull the pen away from the part that's pressed down until the hair straightens out, and draw around.

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    $\begingroup$ The OP says "No additional tools" Hair is used as a tool! $\endgroup$ – DrD Oct 10 at 21:57
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    $\begingroup$ @DrD is right. And that tool is not available to everyone! $\endgroup$ – WhatsUp Oct 10 at 21:58
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    $\begingroup$ I like this answer. Hair is part of your body so it's ok to use it. However the hair needs to be long and strong not to break. $\endgroup$ – Dmitry Kamenetsky Oct 11 at 0:03
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    $\begingroup$ @WhatsUp You have a point. But the accepted answer uses an arm... and not everyone has their arm! $\endgroup$ – ention everyone Oct 11 at 0:33
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    $\begingroup$ Lol ^_^ I would argue that hair can be separated from your body, so that it counts as "tool" while arm is part of yourself :P $\endgroup$ – WhatsUp Oct 11 at 0:43
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Assume the given paper is rectangular. Cut out half of it, wrap it around twice or thrice aligning edges in a plane. Hold it with left hand using slight grip pressure and carefully trace out an almost perfect circle.

Demonstration:

enter image description here

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    $\begingroup$ @risky mysteries Thanks for edit with sketch... $\endgroup$ – Narasimham Oct 11 at 18:18
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if the paper has arbitrary large mass with infinite young's modulus and the friction of the ball point pen writing on the paper is arbitrarily small relative to the papers mass, you can spin the paper in the air at an arbitrary fast speed. applying an arbitrary small force to the pen tip applied to the paper yields a perfect circle that is superior to other methods involving pivots on a compass derived from human fingers or point methods that rely on visual feedback and human motor control.

the trick to solving this problem is that the mass of the paper, the modulus of the paper, the affects of gravity over time, the friction of the pen tip, air friction etc.. are not constrained. this makes any high precision method using physics completely possible and superior to methods that rely on imprecise instruments such as the human hand or the human eye. even if the hand was shaking when the pen tip was applied, the error would not manifest immediately if the spin of the paper was sufficiently much faster than the movement of the pen in non axial trajectory. there would be a moment in time where the circle would be ideal considering the forces used to setup the experiment were ideal. even the affects of axial force with respect to the pen would take time to integrate and act on the paper to change the papers trajectory. therefor all errors are minimized and accounted for. you can for example, apply the pen from the bottom of the paper and use the paper as a gyroscope to completely stabilize the experiment.

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    $\begingroup$ Good luck doing that without any tools! $\endgroup$ – Paul Panzer Oct 11 at 16:39
  • $\begingroup$ the question is obviously flawed with so many variables not being defined or constrained. if the converse were true then any movement you make with your hands would require more force than a human could apply. you would need tools to do anything with the paper or the pen. you could then constrain the human in force and speed so that the human could never finish the experiment. the objection has objections. this illustrates why the question is flawed. $\endgroup$ – acacia Oct 12 at 19:46
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Hold the pen firm with index finger like a scalpel. Press your wrist on the center of the paper, rotate the paper

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