Equation of a Circle Formula

Equation of a circle is the standard form equation (x - h)^2 + (y - k)^2 = r^2 describes a circle with center (h, k) and radius r in the coordinate plane.

The Formula

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2
General form: x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx + Ey + F = 0 (complete the square to convert to standard form).

When to use: A circle is the set of all points at the same distance (the radius) from a center point. The equation just says 'the distance from (x,y)(x, y) to the center (h,k)(h, k) equals rr,' using the distance formula squared.

Quick Example

(x−3)2+(y+1)2=25(x - 3)^2 + (y + 1)^2 = 25 This is a circle with center (3,−1)(3, -1) and radius r=5r = 5.

Notation

Center (h,k)(h, k), radius rr. Note the signs: (x−h)(x - h) means the center's xx-coordinate is +h+h.

What This Formula Means

The standard form equation (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 describes a circle with center (h,k)(h, k) and radius rr in the coordinate plane.

A circle is the set of all points at the same distance (the radius) from a center point. The equation just says 'the distance from (x,y)(x, y) to the center (h,k)(h, k) equals rr,' using the distance formula squared.

Formal View

{(x,y)∈R2∣(x−h)2+(y−k)2=r2}\{(x,y) \in \mathbb{R}^2 \mid (x-h)^2 + (y-k)^2 = r^2\}: the locus of points at distance rr from center (h,k)(h,k)

Worked Examples

Example 1

easy
Write the equation of the circle with center (3,−2)(3, -2) and radius 55.

Answer

(x−3)2+(y+2)2=25(x - 3)^2 + (y + 2)^2 = 25

First step

1
The standard form of a circle's equation is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where (h,k)(h, k) is the center and rr is the radius.

Full solution

  1. 2
    Substitute h=3h = 3, k=−2k = -2, r=5r = 5.
  2. 3
    (x−3)2+(y+2)2=25(x - 3)^2 + (y + 2)^2 = 25.
The equation of a circle is derived from the distance formula: every point (x,y)(x, y) on the circle is exactly rr units from the center (h,k)(h, k). This gives (x−h)2+(y−k)2=r\sqrt{(x-h)^2 + (y-k)^2} = r, which when squared yields the standard form.

Example 2

medium
Find the center and radius of the circle x2+y2−6x+4y−12=0x^2 + y^2 - 6x + 4y - 12 = 0.

Example 3

medium
Convert x2+y2−4x+6y−3=0x^2 + y^2 - 4x + 6y - 3 = 0 to standard form and identify the center and radius.

Common Mistakes

  • Reading the center sign backwards - (x−h)(x-h) gives center +h+h, so (x+2)2(x+2)^2 means center at −2-2.
  • Forgetting to square-root the right side - the equation gives r2r^2, so radius is r2\sqrt{r^2}.
  • Treating it as an ellipse - a circle requires equal coefficients on x2x^2 and y2y^2.

Why This Formula Matters

It is the squared distance formula in disguise and the gateway to all the conics; reading center and radius off the standard form is the single most-tested skill in conic units. The sign trap — (x−h)(x-h) meaning center +h+h — flips half of students' centers if they do not slow down. Recognizing it by "Are x2x^2 and y2y^2 present with equal positive coefficients and a constant on the other side?" — rather than by familiar numbers — is what lets a student tell it apart from ellipse and distance formula and general-form quadratic in x and y in a mixed problem set.

Frequently Asked Questions

What is the Equation of a Circle formula?

The standard form equation (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 describes a circle with center (h,k)(h, k) and radius rr in the coordinate plane.

How do you use the Equation of a Circle formula?

A circle is the set of all points at the same distance (the radius) from a center point. The equation just says 'the distance from (x,y)(x, y) to the center (h,k)(h, k) equals rr,' using the distance formula squared.

What do the symbols mean in the Equation of a Circle formula?

Center (h,k)(h, k), radius rr. Note the signs: (x−h)(x - h) means the center's xx-coordinate is +h+h.

Why is the Equation of a Circle formula important in Math?

It is the squared distance formula in disguise and the gateway to all the conics; reading center and radius off the standard form is the single most-tested skill in conic units. The sign trap — (x−h)(x-h) meaning center +h+h — flips half of students' centers if they do not slow down. Recognizing it by "Are x2x^2 and y2y^2 present with equal positive coefficients and a constant on the other side?" — rather than by familiar numbers — is what lets a student tell it apart from ellipse and distance formula and general-form quadratic in x and y in a mixed problem set.

What do students get wrong about Equation of a Circle?

The procedure for equation of a circle is the easy part; the trap is reading the center sign backwards. Asking "Are x2x^2 and y2y^2 present with equal positive coefficients and a constant on the other side?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Equation of a Circle formula?

Before studying the Equation of a Circle formula, you should understand: pythagorean theorem, domain.

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