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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
There are an infinite number of those points, here are some examples:
x
y
x2 + y2
5
0
52 + 02 = 25 + 0 = 25
3
4
32 + 42 = 9 + 16 = 25
0
5
02 + 52 = 0 + 25 = 25
−4
−3
(−4)2 + (−3)2 = 16 + 9 = 25
0
−5
02 + (−5)2 = 0 + 25 = 25
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
Values we know:22 + y2 = 52
Square root both sides: y = ±√(52 − 22)
y ≈ ±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 = r2
And that is the 'Standard Form' for the equation of a circle!
Things 3 v3 9 2. 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!
Hueshifter 1 2 0 2
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