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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:

x² + y² = 5²

Play fallout 3 windows 10. There are an infinite number of those points, here are some examples:

In all cases a point on the circle follows the rule x² + y² = radius²

We can use that idea to find a missing value

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

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

And here are the two points: Can i install google chrome on my macbook pro.

More General Case

Now let us put the center at (a,b) Starbound 1 4 3 30561 download free.

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 = r²

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 = r²

Put in (a,b) and r:

(x−3)² + (y−4)² = 6²

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

And we end up with this:

x² + y² − 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:

Note: General Form always has x² + y² for the first two terms.

Going From General Form to Standard Form

Now imagine we have an equation in General Form:

x² + y² + Ax + By + C = 0

How can we get it into Standard Form like this?

(x−a)² + (y−b)² = r²

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

Example: x² + y² − 2x − 4y − 4 = 0

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

(x² − 2x + (−1)²) + (y² − 4y) = 4 + (−1)²

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

Sidewriter 1 2 0 Mm

(x² − 2x + (−1)²) + (y² − 4y + (−2)²) = 4 + (−1)² + (−2)²

Tidy up:

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!) Mobisaver 5 0.

Unit Circle

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

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

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The formula for a circle is (x−a)² + (y−b)² = r²

So the center is at (4,2)

And r² 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=':

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.

Sidewriter 1 2 0 2

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