## Introduction: Find the Center of a Circle

This instructable shows how to find the center of a circle. Many, many projects involve circular things and a need to easily find the middle. For various reasons pre-determining the center of a cut circle can be difficult, particularly when it will affect balance in motion.

I was working on the Cardboard Wind Turbine project and the most convenient large diameter template wasn't conveniently sized for pre-measuring and cutting out a square and then matching the template to it seemed tedious and error prone. Also I've never been very good at drawing on the lines. So here we are.

I'd like to thank and acknowledge the folks at Math Open Reference for their contribution.

So what we're going to do is work with a cardboard circle I've cut out, find the center point and then create a "crosshairs" so that work on various projects can proceed smoothly.

## Step 1: Bill of Materials

Here's a list of items we're going to be using.

Carpenters Square or other right angle equal to or larger than the diameter (size) of the circle. I've got a fancy, schmansy carpenters square that I finally found a use for but any reasonably stable right angle will. I've used both the carpenters square and a large manila envelope to demonstrate the utility of the technique.

A compass - I show a basic protractor and compass set available at any drugstore, supermarket or department store. The protractor (the half circle thing) isn't actually used. It is possible to substitute the traditional pencil and string compass but I do not demonstrate that technique. Save yourself a great deal of trouble and spend the two dollars, over time you will thank me.

The box cutter and cardboard box isn't required unless you don't have a circle handy to practice on.

I show two cardboard circles of approximately equal size that I need to find the centers of so I place them on an axle and have them rotate at fairly high speeds so the better the balance the more stable my contraption. Any circle may be substituted.

## Step 2: Finding the Center of the Circle

So how do we find the center? Our solution to this problem lies in a basic law of geometry called Thale's Theorum what this says is that any line of drawn across the diamater a circle always includes a right angle to some point on the circle

So if we place a right angle on the edge of the circle and we mark the spots at which the angle intersects the edge we will have identified a line across the diameter of the circle.

Place the right angle on any point on the edge of the circle so that the legs extend over the edge.

Mark the exit points on both legs of the right angle with a line the edge until it exits the circle.

Use the ruler to draw a line that connects the two points at which the lines exit the circle. This is a diamater line.

Now repeat this process for any other point on the edge of the circle (do not attempt to find the 'right' angle to make the crosshairs, in the next step we will divide the circle by the other half).

The point at which the two lines intersect mark the center of the circle. If that's all you need to find then we're done. Otherwise in the next step we will use our compass to create the classical 'cross hairs' that often needed for further work involving circles or cylinders.

If we do this for a second line the point at which the two diameter lines intersect is the middle.

## Step 3: Creating the Crosshairs

Now that we've found the center in one direction we're going to identify the center in the other direction.

First we have to pick one of our diameter lines to serve as the X axis. We will construct a perpendicular line from the center to create the Y axis and extend it to both sides.

So we're going to take our compass and set it to approximately one half the radius of the circle (or about halfway between the center and the edge).

Placing the point of the compass on the center we going to mark the point at which the compass interects the X axis.

Next we're going to place the point of the compass on each of these and mark where the two circles overlap. It is not necessary to draw the entire circle (or even a very big arc). Just big enough to detect where the two circles would intersect.

Now using the ruler or other straightedge we will draw a line between the intersection and the middle. This is the Y axis line.

Mark the X and Y axis so you don't forget and you are ready to proceed with your project.

## 20 Comments

11 years ago on Introduction

my uncle is a geometry teacher and he taught me this.....im goin to geometry next year! oh yeah! head start!

Reply 11 years ago on Introduction

Math is divided into two great fields, arithmetic (the study of numbers) and geometry (the study of shapes). While all that is living derives from the forms of physics bound by arithmetic all that is beautiful to eye or ear (art and music) derives from geometry. And I will leave you one other tip that will serve you well as you go forward: In your conversations with others remember that little people talk about people, mediocre people talk about things, great people talk about ideas. But don't be a jerk about it....it costs nothing to be gracious

Reply 11 years ago on Introduction

I was kind of being sarcastic about the "head start" thing.......It wasnt my intention to be a jerk.......I went through alebrag 1 and it was alot less simple than i thought it was.....so assuming that geometry is anything close to algebra 1 i think im in for alot more than finding the center of a circle. Also, I dont like talking about people unless i know them personally and know what they go through.

Reply 11 years ago on Introduction

You weren't being a jerk, that was just an observation in general. I'm confident that you'll do well.

Reply 11 years ago on Introduction

just something you felt like saying.....something esoteric? Thanks.

Reply 11 years ago on Introduction

Exactly, while not quite a geezer I'm certainly of an age where I feel completely confident giving unsolicited advice to young people. Having people listen is one of the two greatest things about being grown up. The other is the ability to walk go out for dinner and order dessert first. Here is something I've tried to live by: A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects. -Robert A. Heinlein

Reply 8 years ago on Introduction

"... not quite a geezer I'm certainly of an age where I feel completely confident giving unsolicited advice to young people." Ah ha: you're 52, like me!

Reply 11 years ago on Introduction

Im not much of a dessert person but i like the first one! Id say im smarter than the average bear...not a genius (not even close) and every time i have an idea i share it....NOBODY listens! then when im right i get to rub it in...thats the only good thing about having no one listen to my advice. Also I like that poem..or quote thingy 12 out of 21 aint bad. Well it depends on the definition of "program" is. I like your advice. P.S. I finally figured out how to pronounce your name!

Reply 8 years ago on Introduction

Good going Conrad - what haven't you done yet? These could form part of your "Bucket List". It's 15 out of 21 for me if you substitute chicken or sheep for hog. I don't like fighting so I can't do "invasion" or "fight" (except verbally: I've always said that the best defense is a good offence) and I've not had the chance to die gallantly yet. I've yet to write a sonnet (how about a poem?), set a bone, or comfort the dying. Does leading a gang of renegade Boy Scouts through a forest with camp-made spears count as an "invasion"? That would be 16... :]

Reply 8 years ago on Introduction

Man I was an a-hole in 8th grade. Sorry about all that.

8 years ago on Step 3

Or do it this way :

1. Draw a square around the circle, the length of side of the square being equal to the circle's diameter.

2. Draw a line from top-left to bottom-right corner of square.

3. Draw a line from top-right to bottom-left corner of square.

4. Where the lines cross is the center.

11 years ago on Introduction

I just use a v block and a straight edge. Maybe I should do an insturt on it....

But thanks for this. I was looking for it cause some of the stuff I work on don't fit in a v block.... OMG! I just thought of another idea.egbertfitzwilly you always seem to inspire me!

Reply 11 years ago on Introduction

Thank you for your kind words.

I encourage everyone to write an instructable and disseminate their information. Its impossible to tell how it will it cascade out.

12 years ago on Introduction

Wait...no. Draw to lines of equal lengths (and are shorter than the diameter of the c

Reply 12 years ago on Introduction

This seems to be cut off, did you change your mind about commenting?

12 years ago on Introduction

Fantastic! Now it is necessary to design a small tool that makes it, quickly and reliable.

Reply 12 years ago on Introduction

Like one of these? http://tinyurl.com/dzy8bb

Reply 12 years ago on Introduction

Thanks, very much! That gadget is CEFU, all that I like: Cheap (I can make it), Easy, Fast and Useful. If you trace with that device three or more lines that form a small polygon in the center, you do not need to worry about the thickness of the pencil.

12 years ago on Introduction

... or one of these (slightly different):

http://www.leevalley.com/wood/page.aspx?c=2&p=43205&cat=1,330,49237

12 years ago on Introduction

Thank you very much. I have wondered how to do that, but never got around to looking it up. BTW, the tool actually does it in a different way by drawing the two tangents to the circle from a point, then bisecting the (90 degree) angle.