Equation of a Circle

Standard form equation of a Circle

The standard form equation of a circle is a way to express the definition of a circle on the coordinate plane.

On the coordinate plane, the formula becomes $$(x -h)^2 + (y - k)^2 =r^2 $$

  • h and k are the x and y coordinates of the center of the circle
  • $$(x-9)^2 + (y-6)^2 =100 $$ is a circle centered at (9,6) with a radius of 10

Examples

General Formula general formula equation of circle Circle with a center of (4,3) and a radius of 5 Example 1 Circle with a center of (2, -1) and a radius of 4 example 2

Definition of Circle

Definition : A circle is the set of all points that are the same distance, r, from a fixed point.

General Formula: X 2 + Y 2=r2 where r is the radius

  • Unlike parabolas, circles ALWAYS have X 2 and Y 2 terms.
    X2 + Y2=4 is a circle with a radius of 2 (since 4 =22)
  • Remember that a circle is a locus of points. A circle is all of the points that are a fixed distance, known as the radius, from a given point, known as the center of the circle.
  • Explore the standard equation of a circle using the applet below (Go here for a larger version)

Practice Problems

Practice 1

What is the equation of the circle pictured on the graph below?

Picture of Equation of circle in standard form

Since the radius of this this circle is 1, and its center is the origin, this picture's equation is

$$ (y-0)^2 + (x-0)^2 = 1^2 \\ y^2 + x^2 = 1 $$

Practice 2

Look at the graph below, can you express the equation of the circle in standard form?

Picture of Equation of circle in standard form

Since the radius of this this circle is 1, and its center is (1,0) , this circle's equation is

$$ (y-0)^2 +(x-1)^2 = 1^2 \\ y^2 + (x-1)^2 = 1 $$

Practice 2

Look at the graph below, can you express the equation of the circle in standard form?

Picture of Equation of circle in standard form

Since the radius of this this circle is 2, and its center is (3,1) , this circle's equation is

$$ (x-3)^2 +(y-1)^2 = 2^2 \\ (x-3)^2 +(y-1)^2 = 4 $$

Practice 4

What is the radius of the circles below?

Y2+X2 = 9

$$ \sqrt{9} =3$$

Y2+X2 = 16

$$ \sqrt{16} = 4$$

Y2+X2 = 25

$$ \sqrt{ 25 } = 5 $$

Y2 + X2 = 11

$$ \sqrt{11}$$

Y2 + X2 = a

$$ \sqrt{a}$$

Practice 4

Look at each standard form equation below and identify the center and radius.

r = radius

(y-3)2+(x-1)2 = 9

(1, 3) r = 3

(y-5)2+(x-14)2 = 16

(14, 5) r = 4

(y-1)2+(x-5)2 = 25

(5, 1) r = 5

(x+2)2++(y-12)2 = 36

(-2, 12) r = 6

(y+7)2+(x +5)2 = 49

(-5, -7) r = 7

(x +8)2+(y+17)2 = 49

(-8, -17) r = 7

back to Circles