# Circle Equations

A circle is easy to make:

*Draw a curve that is "radius" away
from a central point.*

And so:

All points are the same distance

from the center.

In fact **the definition** of a circle is

**Circle:** The set of all points on a plane that are a fixed distance from a center.

## Circle on a Graph

Let us put a circle of radius 5 on a graph:

Now let's work out **exactly** where all the points are.

We make a right-angled triangle:

And then use Pythagoras:

x^{2} + y^{2} = 5^{2}

There are an infinite number of those points, here are some examples:

x | y | x^{2} + y^{2} |
---|---|---|

5 | 0 | 5^{2} + 0^{2} = 25 + 0 = 25 |

3 | 4 | 3^{2} + 4^{2} = 9 + 16 = 25 |

0 | 5 | 0^{2} + 5^{2} = 0 + 25 = 25 |

−4 | −3 | (−4)^{2} + (−3)^{2} = 16 + 9 = 25 |

0 | −5 | 0^{2} + (−5)^{2} = 0 + 25 = 25 |

In all cases a point on the circle follows the rule x^{2} + y^{2} = radius^{2}

We can use that idea to find a missing value

### Example: **x** value of 2, and a **radius** of 5

^{2}+ y

^{2}= r

^{2}

^{2}+ y

^{2}= 5

^{2}

^{2}= 5

^{2}− 2

^{2}

^{2}− 2

^{2})

**±4.58...**

*(The ± means there are two possible values: one with + the other with −)*

And here are the two points:

## More General Case

Now let us put the center at **(a,b)**

So the circle is **all the points (x,y)** that are **"r"** away from the center **(a,b)**.

Now lets work out where the points are (using a right-angled triangle and Pythagoras):

It is the same idea as before, but we need to subtract **a** and **b**:

(x−a)^{2} + (y−b)^{2} = r^{2}

And that is the **"Standard Form"** for the equation of a circle!

It shows all the important information at a glance: the center **(a,b)** and the radius **r**.

### Example: A circle with center at (3,4) and a radius of 6:

Start with:

(x−a)^{2} + (y−b)^{2} = r^{2}

Put in (a,b) and r:

(x−3)^{2} + (y−4)^{2} = 6^{2}

We can then use our algebra skills to simplify and rearrange that equation, depending on what we need it for.

## Try it Yourself

## "General Form"

But you may see a circle equation and **not know it**!

Because it may not be in the neat "Standard Form" above.

As an example, let us put some values to a, b and r and then expand it

^{2}+ (y−b)

^{2}= r

^{2}

^{2}+ (y−2)

^{2}= 3

^{2}

^{2}− 2x + 1 + y

^{2}− 4y + 4 = 9

^{2}+ y

^{2}− 2x − 4y + 1 + 4 − 9 = 0

And we end up with this:

x^{2} + y^{2} − 2x − 4y − 4 = 0

It is a circle equation, but "in disguise"!

So when you see something like that think *"hmm ... that might be a circle!"*

In fact we can write it in **"General Form"** by putting constants instead of the numbers:

x^{2} + y^{2} + Ax + By + C = 0

*Note: General Form always has x ^{2} + y^{2} for the first two terms*.

## Going From General Form to Standard Form

Now imagine we have an equation in **General Form**:

x^{2} + y^{2} + Ax + By + C = 0

How can we get it into **Standard Form** like this?

(x−a)^{2} + (y−b)^{2} = r^{2}

The answer is to Complete the Square (read about that) twice ... once for **x** and once for **y**:

### Example: x^{2} + y^{2} − 2x − 4y − 4 = 0

^{2}+ y

^{2}− 2x − 4y − 4 = 0

**x**s and

**y**s together:(x

^{2}− 2x) + (y

^{2}− 4y) − 4 = 0

^{2}− 2x) + (y

^{2}− 4y) = 4

Now complete the square for **x** (take half of the −2, square it, and add to both sides):

(x^{2} − 2x + (−1)^{2}) + (y^{2} − 4y) = 4 + (−1)^{2}

And complete the square for **y** (take half of the −4, square it, and add to both sides):

(x^{2} − 2x + (−1)^{2}) + (y^{2} − 4y + (−2)^{2}) = 4 + (−1)^{2} + (−2)^{2}

Tidy up:

^{2}− 2x + 1) + (y

^{2}− 4y + 4) = 9

^{2}+ (y − 2)

^{2}= 3

^{2}

And we have it in Standard Form!

(Note: this used the a=1, b=2, r=3 example from before, so we got it right!)

## Unit Circle

If we place the circle center at (0,0) and set the radius to 1 we get:

(x−a) (x−0) x |

## How to Plot a Circle by Hand

1. Plot the center **(a,b)**

2. Plot 4 points "radius" away from the center in the up, down, left and right direction

3. Sketch it in!

### Example: Plot (x−4)^{2} + (y−2)^{2} = 25

The formula for a circle is (x−a)^{2} + (y−b)^{2} = r^{2}

So the center is at (4,2)

And **r ^{2}** is

**25**, so the radius is √25 = 5

So we can plot:

- The Center: (4,2)
- Up: (4,2+5) = (4,7)
- Down: (4,2−5) = (4,−3)
- Left: (4−5,2) = (−1,2)
- Right: (4+5,2) = (9,2)

Now, just sketch in the circle the best we can!

## How to Plot a Circle on the Computer

We need to rearrange the formula so we get "y=".

We should end up with two equations (top and bottom of circle) that can then be plotted.

### Example: Plot (x−4)^{2} + (y−2)^{2} = 25

So the center is at (4,2), and the radius is √25 = 5

Rearrange to get "y=":

^{2}+ (y−2)

^{2}= 25

^{2}to the right: (y−2)

^{2}= 25 − (x−4)

^{2}

^{2}]

*(notice the ± "plus/minus" ...*

there can be two square roots!)

there can be two square roots!)

^{2}]

So when we plot these two equations we should have a circle:

- y = 2 + √[25 − (x−4)
^{2}] - y = 2 − √[25 − (x−4)
^{2}]

Try plotting those functions on the Function Grapher.

It is also possible to use the Equation Grapher to do it all in one go.