Circle Calculator
Enter the center (h, k) and radius r. The calculator returns the standard equation (x − h)² + (y − k)² = r², the expanded general form, and the diameter, circumference and area of the circle.
One circle, described five ways
A circle is the set of points at a fixed distance from a fixed centre. Written algebraically that definition becomes (x − h)² + (y − k)² = r², which is simply the distance formula with the distance pinned at r and both sides squared to remove the root.
Give this calculator a centre and a radius and it returns that standard equation, the same circle rearranged into general form, and the three measurements that follow from the radius alone: diameter, circumference and area.
The two equations carry the same information but reveal different things. Standard form shows you the centre and radius at a glance, which is what you want for graphing or for describing a region. General form, x² + y² + Dx + Ey + F = 0, is the shape a problem usually hands you, and it hides both.
Seeing the two side by side is the point. Going from general form back to standard requires completing the square twice; running the conversion forwards here gives you a reference to check that work against.
How to use this calculator
- Enter the centre coordinates h and k h is the horizontal position of the centre, k the vertical. Both may be negative or zero. Note the sign convention: a centre at (−2, −3) produces (x + 2)² + (y + 3)² because the formula subtracts h and k.
- Enter the radius r It must be strictly positive. Zero and negative values are rejected rather than accepted, because neither describes a circle — a radius of zero collapses the figure to a single point.
- Read the standard equation for graphing This is the line to use when plotting by hand or describing the circle to someone else, since the centre and radius are both readable directly from it.
- Read the expanded equation to compare with a textbook problem The general form is what you would be given in a question that asks you to identify the circle, so it is the line to match against your starting point.
The formula, and where it comes from
(x − h)² + (y − k)² = r² d = 2r C = 2πr A = πr²
The standard equation is the Pythagorean distance from (x, y) to (h, k) set equal to r and squared. Squaring is what keeps the equation polynomial and is also why the right-hand side displays r² rather than r — a circle of radius 3 shows 9.
The expansion is done by multiplying out both squares and collecting terms, which gives x² + y² − 2hx − 2ky + (h² + k² − r²) = 0. So the coefficient of x is −2h, the coefficient of y is −2k, and the constant carries the radius information. That is the identity you invert when completing the square: halve the x coefficient and negate it to recover h.
The three measurements use π directly from the language's own constant, so the circumference and area are as accurate as double precision allows before being rounded for display.
What each input means
- h Centre x-coordinate — form field “Center x (h)”
- Horizontal position of the centre. It appears in the standard equation as (x − h), so a positive h shows as a subtraction and a negative h as an addition.
- k Centre y-coordinate — form field “Center y (k)”
- Vertical position of the centre, following the same sign convention as h.
- r Radius — form field “Radius r”
- Distance from the centre to any point on the circle. Must be positive; the solver rejects zero and negatives with an explicit message rather than returning a degenerate figure. Units: same length unit as the coordinates.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
The unit circle
Centre at the origin with radius 1 — the circle underpinning the whole of trigonometry, and the simplest possible check that the conversions are behaving.
Inputs Center x (h) = 0, Center y (k) = 0, Radius r = 1
- Center (h, k) = (0, 0)
- Radius r = 1
- Standard equation (x − 0)² + (y − 0)² = 1
- Expanded equation x² + y² + 0x + 0y -1 = 0
- Diameter d = 2r = 2
- Circumference C = 2πr = 6.28319
- Area A = πr² = 3.14159
Result (x − 0)² + (y − 0)² = 1
With the centre at the origin both linear coefficients vanish, and the general form reduces to x² + y² − 1 = 0. That is the cleanest illustration of what those coefficients encode: they exist only to record how far the centre has moved from the origin.
The circumference and area are numerically 2π and π, reported as decimals rather than as multiples of π. The tool computes numerically throughout and does not produce symbolic output.
A negative centre, and the sign trap
Centre at (−2, −3) with radius 5. Negative coordinates are where the sign convention in the standard form catches people out.
Inputs Center x (h) = -2, Center y (k) = -3, Radius r = 5
- Center (h, k) = (-2, -3)
- Radius r = 5
- Standard equation (x + 2)² + (y + 3)² = 25
- Expanded equation x² + y² + 4x + 6y -12 = 0
- Diameter d = 2r = 10
- Circumference C = 2πr = 31.4159
- Area A = πr² = 78.5398
Result (x + 2)² + (y + 3)² = 25
The standard equation shows plus signs inside both brackets, because subtracting a negative adds. Reading (x + 2)² backwards and concluding the centre is at +2 is the single most common error with this form; the centre line above the equation is there to settle it.
In the expanded form the coefficients come out as +4x and +6y — that is −2h and −2k with h and k negative. Halving and negating recovers the centre, which is the manual check worth doing when converting in the other direction.
Reading the result
The right-hand side is r², not r
A circle of radius 3 displays 9 on the right of the standard equation. Reading that number as the radius is a frequent slip; take its square root first. Conversely, a right-hand side that is not a perfect square simply means the radius is irrational, which is perfectly ordinary.
Reading the centre off the expanded form
Given x² + y² + Dx + Ey + F = 0, the centre is at (−D/2, −E/2) and the radius satisfies r² = D²/4 + E²/4 − F. If that expression comes out negative there is no real circle, which is why not every general quadratic of this shape describes one.
Precision of the measurements
Diameter is exact whenever the radius is. Circumference and area involve π, so they are irrational for any rational radius and are shown to six significant figures.
When you would use this
Converting between the two equation forms
Coursework routinely asks for the centre and radius of a circle given in general form. Entering your recovered centre and radius here and comparing the expanded line against the original equation confirms the completing-the-square work in one step.
Laying out circular geometry
For a circular path, a rotating part or a coverage radius on a map, the centre and radius are known and the equation, circumference and area are what you need.
Assumptions and limitations
What this calculator assumes
- The radius is strictly positive; zero and negative radii are rejected rather than interpreted.
- Coordinates and radius share one unit — the tool performs no unit conversion, so the area is in that unit squared.
Where it stops being the right tool
- Circles only. Ellipses, where the two axes differ, need the ellipse tool instead.
- One direction of conversion: enter centre and radius to get both equations. Entering a general-form equation to recover the centre is not supported.
Common mistakes
Reading the centre with the wrong sign
Why it happens. The standard form subtracts h and k, so (x + 2)² corresponds to h = −2. The bracket shows the opposite sign to the coordinate, and under time pressure the visible sign wins.
How to avoid it. Rewrite the bracket as a subtraction before reading it: (x + 2) is (x − (−2)), so the centre coordinate is −2. The centre line in the output confirms it independently.
Taking the right-hand side as the radius
Why it happens. The number sits alone on the right of an equation whose subject looks like a distance, so it reads as one. It is the square of the radius.
How to avoid it. Take the square root. If the equation ends in 25, the radius is 5 — and the diameter line in the output gives an independent cross-check.
Halving the x coefficient without negating it
Why it happens. Recovering the centre from general form needs both operations, and the negation is easy to drop because the halving feels like the substantive step.
How to avoid it. The centre is (−D/2, −E/2). Test it here: enter your recovered centre and check the expanded line reproduces the coefficients you started from.
Frequently asked questions
What is the standard equation of a circle?
(x − h)² + (y − k)² = r², where (h, k) is the centre and r the radius. It is the distance formula with the distance fixed at r and both sides squared, which is why the right-hand side shows r² rather than r.
How do standard and expanded form differ?
Standard form shows the centre and radius directly and is what you want for graphing. Expanded form, x² + y² + Dx + Ey + F = 0, is what a problem usually hands you and hides both — recovering them means completing the square in x and in y.
Why is the radius required to be positive?
A radius of zero collapses the circle to a single point and a negative radius has no geometric meaning, so neither describes a circle. The solver rejects both with a message rather than returning a degenerate figure.
Why are the circumference and area decimals rather than multiples of π?
The calculation is numeric throughout and uses the language's own value of π, so results are reported to six significant figures. For an exact form, divide the reported circumference by π and look for a simple value.