Equation Of A Circle Calculator

What this calculator does

On a graph, a circle can be written down as an equation, and there is more than one way to write it. This tool takes the form you have, converts it into the other forms, and pulls out the circle's centre, radius, area, and perimeter along the way.

It is the coordinate-geometry companion to the plain circle tools: instead of measuring a circle, you describe where it sits on the plane and how big it is.

Using the calculator

1. Pick the equation type you are starting from: general, standard, or parametric. The input boxes change to match.
2. Type the values for that form.
3. Set the precision (how many decimal places) if you want something other than two, then press Calculate.

You get the centre and radius, the area and perimeter, and the circle rewritten in all three forms.

The three forms, and why each exists

They all describe the same circle. They just suit different jobs.

Standard form: (x - A)² + (y - B)² = C. The friendly one. The centre (A, B) and the radius (the square root of C) are sitting right there in the open, so it is the form you want for reading a circle off or sketching it.

General form: x² + y² + Dx + Ey + F = 0. The same circle, multiplied out and tidied up. The centre and radius are now hidden inside the coefficients D, E and F. This is usually the shape a textbook problem hands you, and the job is to dig the circle back out.

Parametric form: x = A + r cos(α), y = B + r sin(α). This one describes the circle as a moving point. As the angle α sweeps from 0 to 360 degrees, the point traces all the way around. It is the form you reach for when plotting a circle or animating something around one.

How the centre and radius are found

Everything the tool reports comes from one pair of facts about the circle: where its centre is, and how big its radius is. Here is how it gets them from each form.

• From standard form (A, B, C): the centre is (A, B), and the radius is √C.
• From general form (D, E, F): the centre is (-D ÷ 2, -E ÷ 2), and the radius is √((D ÷ 2)² + (E ÷ 2)² - F). This is what falls out when you complete the square on the general equation.
• From parametric form (A, B, r): the centre (A, B) and the radius r are already laid bare, nothing to undo.

Once it has the radius, the rest is the usual pair: area = π × radius² and perimeter = 2 × π × radius. The precision box just sets how many decimals the answers are rounded to.

A worked example | general to standard form

Take the general equation x² + y² - 6x + 4y - 12 = 0. So D = -6, E = 4 and F = -12.

1. Centre: (-D ÷ 2, -E ÷ 2) = (3, -2).
2. Radius: √((-3)² + 2² - (-12)) = √(9 + 4 + 12) = √25 = 5.

So the circle is centred at (3, -2) with a radius of 5. Written in standard form that is (x - 3)² + (y + 2)² = 25, and from the radius the area is π × 25 = 78.54 and the perimeter is 2 × π × 5 = 31.42.

When the numbers do not make a circle

There is a quiet check built into the general form. The radius comes from √((D ÷ 2)² + (E ÷ 2)² - F), and a square root only works on a number that is zero or positive. If the part under the root comes out negative, there is no real circle that fits those coefficients, so the tool stops and says so rather than inventing one.

That makes it handy for a common exam question: feed in an equation, and a clean centre and radius means it really was a circle. And if the shape you are working with looks stretched rather than round, it is probably an ellipse, which the area of an ellipse calculator handles.

Questions people ask

How do I find the centre and radius from the general form?

The centre is (-D ÷ 2, -E ÷ 2), and the radius is √((D ÷ 2)² + (E ÷ 2)² - F). It comes from completing the square on x² + y² + Dx + Ey + F = 0.

What is the difference between standard and general form?

Standard form, (x - A)² + (y - B)² = C, shows the centre and radius openly. General form, x² + y² + Dx + Ey + F = 0, is the same circle multiplied out, with those values hidden in the coefficients.

What is the parametric form for?

It describes each point on the circle by an angle: x = A + r cos(α), y = B + r sin(α). As the angle goes from 0 to 360 degrees, the point travels around the whole circle, which is useful for graphing and animation.

What does it mean if there is no circle?

It means the part under the square root in the radius came out negative, so no real circle matches those numbers. The equation you entered was not a valid circle in general form.

Does this need units?

No, you are working with coordinates, so the values are just numbers. If you do attach a unit, remember the area comes out in that unit squared.

References

A note on where this comes from. Writing a circle as an equation on a grid is the heart of analytic geometry, the approach René Descartes set out in La Géométrie in 1637, which married algebra to shapes on a plane. The area and perimeter the tool reports rest on the value of π as tabulated by the US National Institute of Standards and Technology. For further reading, see Circle.

1. René Descartes, La Géométrie (1637), the founding work of analytic geometry, where shapes are described by equations on a coordinate plane.
2. National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, value of π. https://dlmf.nist.gov/