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 = πr2

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:

x2 + y2 = r2

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 = r2

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:

x2 + y2 = r2


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 = 52

(x - 4)2 + (y + 3)2 = 25

General form of a circle

The equation of a circle in general form is,

x2 + y2 + 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.


Find the center and radius for the circle with equation,

x2 + y2 + 4x - 12y - 9 = 0

Group the x and y terms first.

x2 + 4x + y2 - 12y = 9

Completing the square, we get:

x2 + 4x + 4 + y2 - 12y + 36 = 9 + 4 + 36

(x + 2)2 + (y - 6)2 = 49 = 72

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