# Circle formula

The set of all points in a plane that are equidistant from a fixed point, defined as the center, is called a circle.

Formulas involving circles often contain a mathematical constant, pi, denoted as π; π ≈ 3.14159. π is defined as the ratio of the circumference of a circle to its diameter. Two of the most widely used circle formulas are those for the circumference and area of a circle.

## Circumference formula of a circle

The circumference, C, of a circle is a measure of the distance around the circle. It can be found using the formula

C = 2πr

where r is the circle's radius.

If using the diameter, d, of the circle:

C = πd

If the area of the circle, A, is known:

## Area formula of a circle

The area of a circle is the plane region bounded by the circle's circumference. It can be found using the formula

A = πr^{2}

where r is the circle's radius.

If using the diameter, d, of the circle:

If the circumference of the circle, C, is known:

## Equation of a circle

In coordinate geometry, a circle can be expressed using a number of equations based on various constraints.

### Centered at the origin

Given that point (x, y) lies on a circle with radius r centered at the origin of the coordinate plane, it forms a right triangle with sides x and y, and hypotenuse r. This allows us to use the Pythagorean Theorem to find that the equation for this circle in standard form is:

x^{2} + y^{2} = r^{2}

This is true for any point on the circle since any point on the circle is an equal distance, r, from the center.

### Centered at any location

To find the equation for a circle in the coordinate plane that is not centered at the origin, we use the distance formula. This method can also be used to find the equation for a circle centered at the origin, but in such a case, using the equation in the previous section would be more efficient.

Given a circle with radius, r, centered at point (h, k), we can use the distance formula to find that:

where (x, y) is any point on the circle.

Squaring both sides of the equation, we get the equation of the circle:

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

Notice that if the circle is centered at the origin, (0, 0), then both h and k in the equation above are 0, and the equation reduces to what we got in the previous section:

x^{2} + y^{2} = r^{2}

Example:

Find the equation of the circle with center (4, -3) and radius 5.

Substituting the coordinates of the center and radius we get,

(x - 4)^{2} + (y -( -3))^{2} = 5^{2}

(x - 4)^{2} + (y + 3)^{2} = 25

### General form of a circle

The equation of a circle in general form is,

x^{2} + y^{2} + Dx +Ey + F = 0

where D, E, and F are real numbers.

To more easily identify the center and radius of a circle given in general form, we can convert the equation to standard form.

Example:

Find the center and radius for the circle with equation,

x^{2} + y^{2} + 4x - 12y - 9 = 0

Group the x and y terms first.

x^{2} + 4x + y^{2} - 12y = 9

Completing the square, we get:

x^{2} + 4x + 4 + y^{2} - 12y + 36 = 9 + 4 + 36

(x + 2)^{2} + (y - 6)^{2} = 49 = 7^{2}

So, the center is (-2, 6) and the radius is 7.