Circle equation
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.
Draw a curve that is "radius" away
from a central point.
And so:
All points are the same distance from the center.
Let us put that center at (a,b).
So the circle is all the points (x,y) that are "r" away from the center (a,b).
Now we can work out exactly where all those points are!
We simply make a right-angled triangle (as shown), and then use Pythagoras (a2 + b2 = c2):
So the circle is all the points (x,y) that are "r" away from the center (a,b).
Now we can work out exactly where all those points are!
We simply make a right-angled triangle (as shown), and then use Pythagoras (a2 + b2 = c2):
(x-a)2 + (y-b)2 = r2
And that is the "Standard Form" for
the equation of a circle!
And that is the "Standard Form" for
the equation of a circle!
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
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
- Start with: (x-a)2 + (y-b)2 = r2
- Set (for example) a=1, b=2, c=3: (x-1)2 + (y-2)2 = 32
- Expand: x2 - 2x + 1 + y2 - 4y + 4 = 9
- Gather like terms: x2 + y2 - 2x - 4y + 1 + 4 - 9 = 0
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!"
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!"
Unit Circle
If we place the circle center at (0,0) and set the radius to 1 we get:
(x-a)2 + (y-b)2 = r2
(x-0)2 + (y-0)2 = 12
x2 + y2 = 1
Which is the equation of the Unit Circle?
If we place the circle center at (0,0) and set the radius to 1 we get:
(x-a)2 + (y-b)2 = r2
(x-0)2 + (y-0)2 = 12
x2 + y2 = 1
Which is the equation of the Unit Circle?
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!
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 = 25The 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:
Now, just sketch in the circle the best that you can!
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 that you can!
