Equation of a circle

There is a general equation which defines the shape of a circle on a graph.

For any point that lies on the circle, the equation will be true and this creates a circle if we would plot it using a graphing calculator, for example.

General equation of a circle

Where:

• is the coordinate of the centre of the circle:
  • is the x-coordinate of the centre
  • is the y-coordinate of the centre
• is the radius of the circle.

Equation of a circle at the origin

If we have a circle with a centre at the origin (0,0), then the equation would be , which simplifies to:

Where:

• is the radius

Finding the radius and centre of a circle from its equation

Knowing the equation above, we can use that to find the radius and/or the centre coordinates of a circle, given just its equation.

Example: find the radius of the circle with equation

• Given the equation above, we know that must be the radius squared:
• Answer: the radius is

Example: find the centre of the circle with equation

• For the general form :
  • is
  • is
  • is (so is )
• The coordinates of the centre are .
• and , so
• Answer: the centre is

Example: find the centre of the circle with equation

• For the general form :
  • is : negative, because is equivalent to
  • is
  • is (so is )
• The coordinates of the centre are .
• and , so
• Answer: the centre is

Example: find the centre and radius of the circle with equation

• Circle is at the origin, so:
• Centre at
• Answer: the centre is and the radius is

Finding the equation of a circle from its centre and radius

We can also go the other way round - finding the equation of a circle if we know its centre and its radius.

Example: find the equation of the circle with centre and radius

• Substitute for , and :
• Answer: the equation is

Example: find the equation of the circle with centre and radius

• Substitute for , and :
• Answer: the equation is

Example: find the equation of the circle with centre and radius

• Substitute for , and :
• Answer: the equation is