What is the standard form of a circle equation?

The standard form is (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the centre and r is the radius. Every point (x, y) on the circle is exactly r units from the centre, which is the definition of a circle.

How do I convert the general form to standard form?

The general form is x^2 + y^2 + Dx + Ey + F = 0. Complete the square on x and y: group the x terms, add (D/2)^2 to both sides, group the y terms, add (E/2)^2 to both sides. The result is (x + D/2)^2 + (y + E/2)^2 = D^2/4 + E^2/4 - F, so the centre is (-D/2, -E/2) and r = sqrt(D^2/4 + E^2/4 - F). This calculator does it for you instantly.

What is the formula for arc length?

Arc length L = r * theta, where r is the radius and theta is the central angle in radians. To convert degrees to radians multiply by pi/180. For example a 90-degree arc on a circle of radius 5 has length 5 * (pi/2) which is approximately 7.854 units.

How is chord length calculated?

Chord length = 2r sin(theta/2), where theta is the central angle subtended by the chord. The sagitta — the perpendicular distance from the midpoint of the chord to the arc — equals r(1 - cos(theta/2)). Both are shown in the chord-length mode.

What is a sector, and how is its area calculated?

A sector is the pie-slice region bounded by two radii and the arc between them. Its area equals (1/2) r^2 theta where theta is the central angle in radians, which is the same as (theta/360) * pi * r^2 in degrees. For a full circle theta = 2*pi and the formula reduces to the familiar pi*r^2.

When does the general-form solver return "no real circle"?

When D^2/4 + E^2/4 - F is negative the equation represents an imaginary circle — no real points satisfy it. When it equals zero the equation represents a single point (a degenerate circle of radius zero). Only positive values of that discriminant yield a real circle with a positive radius.

Circle Equation Calculator

A circle equation calculator that handles every common geometry task in one place: write the equation of a circle in standard form or general form, convert between the two, compute area and circumference, find arc length, chord length (plus the sagitta), and sector area — all from the radius and a central angle. A live SVG diagram updates as you type, and a Copy button lets you paste results directly into documents or spreadsheets. Everything runs client-side; no data leaves your browser.

How it works

A circle is the set of all points in a plane that are the same distance — the radius — from a fixed point called the centre. That single idea generates every formula below.

Standard form

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

Here (h, k) is the centre and r is the radius. The equation simply says “the distance from (x, y) to (h, k) equals r”, written via the Pythagorean theorem. This is the most readable form: you can read off the centre and radius by inspection.

General form

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

Expanding the standard form and collecting terms gives the general form. The relationships are D = -2h, E = -2k and F = h^2 + k^2 - r^2, so:

Centre = (-D/2, -E/2)
r^2 = D^2/4 + E^2/4 - F

The calculator recovers both when you enter D, E and F.

Area and circumference

Quantity	Formula
Area	pi * r^2
Circumference	2 * pi * r
Diameter	2r

Arc length and sector area

For a central angle theta (in radians):

Arc length L = r * theta
Sector area A = (1/2) * r^2 * theta

The calculator accepts degrees and converts internally (theta_rad = theta_deg * pi / 180).

Chord length and sagitta

A chord connects two points on the circle. If the central angle subtended is theta:

Chord length = 2r sin(theta/2)
Sagitta h = r(1 - cos(theta/2))

The sagitta is the height of the circular segment — the bulge between the chord and the arc.

Worked example

A circle has centre (3, -2) and radius 5.

Standard form: (x - 3)^2 + (y + 2)^2 = 25

General form: expand and collect: x^2 - 6x + 9 + y^2 + 4y + 4 = 25, so x^2 + y^2 - 6x + 4y - 12 = 0 (D = -6, E = 4, F = -12)

Measurements:

Area = pi * 25 ≈ 78.54 square units
Circumference = 10*pi ≈ 31.42 units

A 60-degree arc on this circle:

theta_rad = pi/3 ≈ 1.0472
Arc length = 5 * pi/3 ≈ 5.236 units
Chord length = 2 * 5 * sin(30°) = 10 * 0.5 = 5 units exactly
Sagitta = 5(1 - cos(30°)) ≈ 5(1 - 0.866) ≈ 0.67 units
Sector area = (1/2) * 25 * pi/3 ≈ 13.09 square units

All of these numbers appear automatically when you enter r = 5 and angle = 60 in the relevant mode of the calculator.

Formula reference

Quantity	Formula
Standard form	(x-h)^2 + (y-k)^2 = r^2
General form	x^2 + y^2 + Dx + Ey + F = 0
Centre from general	(-D/2, -E/2)
Radius from general	sqrt(D^2/4 + E^2/4 - F)
Area	pi*r^2
Circumference	2pir
Arc length	r * theta (theta in radians)
Chord length	2r * sin(theta/2)
Sagitta	r*(1 - cos(theta/2))
Sector area	(1/2)r^2theta

Units are consistent throughout: lengths and areas share whichever unit you use for the radius (metres, centimetres, inches, etc.). The calculator never sends data to any server.