12.2 Circles

The definition of a circle

You’ve known all your life what a circle looks like. You probably know how to find the area and the circumference of a circle, given its radius. But what is the exact mathematical definition of a circle? Before you read the answer, you may want to think about the question for a minute. Try to think of a precise, specific definition of exactly what a circle is.

Below is the definition mathematicians use.

Definition of a circle

Mathematicians often seem to be deliberately obscuring things by creating complicated definitions for things you already understood anyway. But if you try to find a simpler definition of exactly what a circle is, you will be surprised at how difficult it is. Most people start with something like “a shape that is round all the way around.” That does describe a circle, but it also describes many other shapes, such as this pretzel:

So you start adding caveats like “it can’t cross itself” and “it can’t have any loose ends.” And then somebody draws an egg shape that fits all your criteria, and yet is still not a circle:

So you try to modify your definition further to exclude that ... and by that time, the mathematician’s definition is starting to look beautifully simple.

But does that original definition actually produce a circle? The following experiment is one of the best ways to convince yourself that it does.

Experiment: drawing the perfect circle

1. Lay a piece of cardboard on the floor.
2. Thumbtack one end of a string to the cardboard.
3. Tie the other end of the string to your pen.
4. Pull the string as tight as you can, and then put the pen on the cardboard.
5. Pull the pen all the way around the thumbtack, keeping the string taut at all times.

The pen will touch every point on the cardboard that is exactly one string-length away from the thumbtack. And the resulting shape will be a circle. The cardboard is the plane in our definition, the thumbtack is the center , and the string length is the radius .

The purpose of this experiment is to convince yourself that if you take all the points in a plane that are a given distance from a given point, the result is a circle. We’ll come back to this definition shortly, to clarify it and to show how it connects to the mathematical formula for a circle.

The mathematical formula for a circle

You already know the formula for a line: $y=mx+b$ . You know that $m$ is the slope, and $b$ is the y-intercept. Knowing all this, you can easily answer questions such as: “Draw the graph of $y=2x\mathrm{–3}$ ” or “Find the equation of a line that contains the points (3,5) and (4,4).” If you are given the equation $3x+2y=6$ , you know how to graph it in two steps: first put it in the standard $y=mx+b$ form, and then graph it.

All the conic sections are graphed in a similar way. There is a standard form which is very easy to graph, once you understand what all the parts mean. If you are given an equation that is not in standard form, you put it into the standard form, and then graph it.