Equation of a Circle Formula

Equation of a Circle: 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
General form: x2+y2+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) to the center (h,k) equals r,' using the distance formula squared.

Quick Example

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

Notation

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

What This Formula Means

The standard form equation (x−h)2+(y−k)2=r2 describes a circle with center (h,k) and radius r 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) to the center (h,k) equals r,' using the distance formula squared.

Formal View

{(x,y)∈R2∣(x−h)2+(y−k)2=r2}: the locus of points at distance r from center (h,k)

Worked Examples

Example 1

easy
Write the equation of the circle with center (3,−2) and radius 5.

Answer

(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, where (h,k) is the center and r is the radius.

Full solution

  1. 2
    Substitute h=3, k=−2, r=5.
  2. 3
    (x−3)2+(y+2)2=25.
The equation of a circle is derived from the distance formula: every point (x,y) on the circle is exactly r units from the center (h,k). This gives (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=0.

Example 3

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

Common Mistakes

  • Reading the center sign backwards - (x−h) gives center +h, so (x+2)2 means center at −2.
  • Forgetting to square-root the right side - the equation gives r2, so radius is r2.
  • Treating it as an ellipse - a circle requires equal coefficients on x2 and y2.

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) meaning center +h — flips half of students' centers if they do not slow down. Recognizing it by "Are x2 and y2 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 describes a circle with center (h,k) and radius r 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) to the center (h,k) equals r,' using the distance formula squared.

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

Center (h,k), radius r. Note the signs: (x−h) means the center's x-coordinate is +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) meaning center +h — flips half of students' centers if they do not slow down. Recognizing it by "Are x2 and y2 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 x2 and y2 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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