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

(x-a)^2+(y-b)^2=r^2 $

Where:

• (a, b) is the coordinate of the centre of the circle:
  • a is the x-coordinate of the centre
  • b is the y-coordinate of the centre
• r 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 (x-0)^2+(y-0)^2=r^2, which simplifies to:

x^2+y^2=r^2

Where:

• r 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 (x-4)^2+(x+2)^2=36

• Given the equation above, we know that 36 must be the radius squared:
• r^2=36
• r=\sqrt{36}=6
• Answer: the radius is 6

Example: find the centre of the circle with equation (x-3)^2+(x-2)^2=16

• For the general form (x-a)^2+(y-b)^2=r^2:
  • a is 3
  • b is 2
  • r^2 is 4 (so r is 4)
• The coordinates of the centre are (a,b).
• a=3 and b=2, so (a,b)=(3,2)
• Answer: the centre is (3,2)

Example: find the centre of the circle with equation (x+3)^2+(x-2)^2=16

• For the general form (x-a)^2+(y-b)^2=r^2:
  • a is -3: negative, because x+3 is equivalent to x-(-3)
  • b is 2
  • r^2 is 4 (so r is 4)
• The coordinates of the centre are (a,b).
• a=-3 and b=2, so (a,b)=(-3,2)
• Answer: the centre is (-3,2)

Example: find the centre and radius of the circle with equation x^2+y^2=25

• Circle is at the origin, so:
  • a=0
  • b=0
• Centre at (a,b)=(0,0)
• r^2=25
• r=\sqrt{25}=5
• Answer: the centre is (0,0) and the radius is 5

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 (4,5) and radius 7

• (a,b)=(4,5)
• a=4
• b=5
• r=7
• (x-a)^2+(x-b)^2=r^2
• Substitute for a, b and r:
  • (x-4)^2+(y-5)^2=7^2
  • (x-4)^2+(y-5)^2=49
• Answer: the equation is (x-4)^2+(y-5)^2=49

Example: find the equation of the circle with centre (-3,2) and radius 5

• (a,b)=(-3,2)
• a=-3
• b=2
• r=5
• (x-a)^2+(x-b)^2=r^2
• Substitute for a, b and r:
  • (x-(-3))^2+(y-2)^2=5^2
  • (x+3)^2+(y-2)^2=25
• Answer: the equation is (x+3)^2+(y-2)^2=25

Example: find the equation of the circle with centre (0,0) and radius 4

• (a,b)=(0,0)
• a=0
• b=0
• r=4
• (x-a)^2+(x-b)^2=r^2
• Substitute for a, b and r:
  • (x-0)^2+(y-0)^2=4^2
  • x^2+y^2=16
• Answer: the equation is x^2+y^2=16