# 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 Circle with a center of (4,3) and a radius of 5

### Another Example

Circle with a center of (2, -1) and a radius of 4

### 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

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

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

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

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

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

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

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

$$\sqrt{a}$$

(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