# Circle: Equation

How to find and use the equation of a circle: formula, 4 examples, and their solutions.

## Formula

If the center of a circle is (h, k)

and if the radius is r,

then the equation of the circle is

(x - h)^{2} + (y - k)^{2} = r^{2}.

## ExampleCenter: (2, 1), r = 3

The center of the circle is (2, 1).

r = 3

Then the equation of the circle is

(x - 2)^{2} + (y - 1)^{2} = 3^{2}.

3^{2} = 9

So

(x - 2)^{2} + (y - 1)^{2} = 9

is the answer.

## ExampleCenter: (4, 0), Diameter: 14

The center of the circle is (4, 0).

The diameter of the circle is 14.

So the radius r is, 14/2, 7.

Center: (4, 0)

r = 7

Then the equation of the circle is

(x - 4)^{2} + (y - 0)^{2} = 7^{2}.

+(y - 0)^{2} = +y^{2}

7^{2} = 49

So

(x - 4)^{2} + y^{2} = 49

is the answer.

## ExampleEndpoints: (-1, 3), (7, 1)

First draw the condition.

Draw a circle.

Draw the diameter.

And draw the endpoints of the diameter

(-1, 3) and (7, 1).

The midpoint of the diameter M

is the center of the circle.

And the divided segments

are the r (radius).

Find the center M.

M is the midpoint of (-1, 3) and (7, 1).

So M([-1 + 7]/2, [3 + 1]/2).

-1 + 7 = 6

3 + 1 = 4

6/2 = 3

4/2 = 2

So M(3, 2).

Write (3, 2)

on the center of the circle.

Find the radius r.

r is the distance between

the center M(3, 2) and the endpoint (7, 1).

So r = √(7 - 3)^{2} + (1 - 2)^{2}

Distance Formula

You can also choose

M(3, 2) and the other endpoint (-1, 3).

You'll get the same answer.

7 - 3 = 4

1 - 2 = -1

4^{2} = 16

+(-1)^{2} = +1

16 + 1 = 17

So r = √17.

Write √17

on the radius r of the circle.

Center: (3, 2)

r = √17

Then the equation of the circle is

(x - 3)^{2} + (y - 2)^{2} = (√17)^{2}.

(√17)^{2} = 17

Square Root

So

(x - 3)^{2} + (y - 2)^{2} = 17

is the answer.

## ExampleGeneral Form → Center, Radius

The given circle is in general form:

x^{2} + y^{2} + Ax + By + C = 0.

To find the center and the r,

change this equation to standard form:

(x - h)^{2} + (y - k)^{2} = r^{2}.

First, move the constant term +20

to the right side.

Use x^{2} - 4x

to make a perfect square trinomial.

x^{2} is x^{2}.

-4x is

-2 times

x times,

(-4x)/(-2⋅x), 2.

Write +2^{2}.

Quadratic Equation: Completing the Square

Use +y^{2} + 10y

to make a perfect square trinomial.

y^{2} is y^{2}.

+10y is

+2 times

y times,

(+10y)/(+2⋅y), 5.

Write +5^{2}.

Write the right side -20.

And to undo +2^{2} and +5^{2},

write +2^{2} + 5^{2}

on the right side.

x^{2} - 2⋅x⋅2 + 2^{2}

= (x - 2)^{2}

y^{2} + 2⋅y⋅5 + 5^{2}

= (y + 5)^{2}

Factor a Perfect Square Trinomial

+2^{2} = +4

+5^{2} = +25

-20 + 25 = 5

5 + 4 = 9

To find the radius easily,

change 9 to 3^{2}.

To find the y value of the center easily,

change +(y + 5)^{2} to +(y - (-5))^{2}.

Then (x - 2)^{2} + (y - (-5))^{2} = 3^{2}.

See the circle equation

(x - 2)^{2} + (y - (-5))^{2} = 3^{2}.

The center is (2, -5).

The radius is 3.

So

Center: (2, -5)

Radius: 3

is the answer.