Analytic Geometry Calculator

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equation of a circle calculator

Equation of a circle: center form, general form, center and radius

In the Circle tab, choose how to define it: center and radius, diameter endpoints, three points or the equation. The calculator shows the center form and the general form, the center, the radius, the circumference, the area and the drawing.

Completing the square: from the general to the center form

The circle with center C(a, b) and radius r has the center-form equation (x - a)² + (y - b)² = r², which comes from the definition: every point is at distance r from the center. Expanding the squares gives the general equation x² + y² - 2ax - 2by + (a² + b² - r²) = 0, which the calculator also writes. For center (2, 3) and radius 5 the center form is (x - 2)² + (y - 3)² = 25 and the general form is x² + y² - 4x - 6y - 12 = 0, since 4 + 9 - 25 = -12. In the general form the x² and y² coefficients are equal and there is no xy term; without that the curve is not a circle (it may be an ellipse, for example).
To go from the general to the center form, group the x terms and the y terms and add to each group the square of half its coefficient, on both sides. In x² + y² - 4x + 6y - 12 = 0 this gives (x² - 4x + 4) + (y² + 6y + 9) = 12 + 4 + 9, that is, (x - 2)² + (y + 3)² = 25. The center is (2, -3), with the signs flipped relative to the parentheses, and the radius is √25 = 5. If the right-hand side comes out zero, the equation describes only the center point; if negative, no real circle exists, and the calculator tells you which case occurred instead of showing the root of a negative number.
There are other ways to define a circle. By the endpoints of a diameter, the center is the midpoint and r = AB/2: A(1, 2) and B(5, 4) give center (3, 3) and r² = 5, so r = √5. By three non-collinear points, you solve the perpendicular bisectors of two pairs: (0, 0), (4, 0) and (0, 2) give center (2, 1) and (x - 2)² + (y - 1)² = 5. The disk area is πr² and the circumference is 2πr, written exactly, such as 25π and 10π. With a point P entered, the squared distance to the center is compared with r²: smaller is inside, equal is on the curve, larger is outside.

Checked examples

Center and radius

Input: C(2, 3), r = 5
Expected output: (x - 2)² + (y - 3)² = 25; x² + y² - 4x - 6y - 12 = 0

Circumference 10π, area 25π.

General to center form

Input: x² + y² - 4x + 6y - 12 = 0
Expected output: (x - 2)² + (y + 3)² = 25; C(2, -3), r = 5

Added 4 + 9 to both sides.

Another negative center

Input: x² + y² + 2x - 4y - 4 = 0
Expected output: (x + 1)² + (y - 2)² = 9; C(-1, 2), r = 3

4 + 1 + 4 = 9.

From a diameter

Input: A(1, 2), B(5, 4)
Expected output: (x - 3)² + (y - 3)² = 5; r = √5

Circumference 2π√5.

Through three points

Input: (0, 0), (4, 0), (0, 2)
Expected output: (x - 2)² + (y - 1)² = 5

Center (2, 1).

No real circle

Input: x² + y² - 6x - 8y + 30 = 0
Expected output: r² = -5

Negative squared radius.

Full tool FAQ

How do you calculate the distance between two points?

Use d = √((x₂ - x₁)² + (y₂ - y₁)²), which is the Pythagorean theorem applied to the coordinate differences. Between A(1, 2) and B(5, 4): d = √(4² + 2²) = √20 = 2√5 ≈ 4.4721. The calculator simplifies the radical by pulling perfect squares out of the root (20 = 4 · 5) and shows the decimal next to it.

How do you find the equation of the line through two points?

What is the difference between the standard, slope-intercept, intercept and parametric forms?

How do you calculate the slope and the angle of inclination?

How can you tell whether two lines are parallel or perpendicular?

How do you find the intersection of two lines?

How do you calculate the distance from a point to a line?

How do you check whether three points are collinear and find the triangle area?

How do you get the equation of a circle and go from general form back to center form?

Are the results exact? What are the limits?

Frequently asked questions

How do I find the center and radius from the general equation?

In x² + y² + Dx + Ey + F = 0, the center is (-D/2, -E/2) and r² = (D/2)² + (E/2)² - F. If the x² and y² coefficients differ from 1 (but are equal to each other), divide the equation by them first.

How do I know whether a line is secant, tangent or outside?

Compare the distance from the center to the line with the radius. Smaller: the line cuts at two points (secant). Equal: it touches at one point (tangent). Larger: it does not touch. The calculator shows the exact intersection points, with radicals, and the chord length.

Are the circle and the disk the same thing?

In coordinate geometry, the circle is the curve (x - a)² + (y - b)² = r², and the disk is the region with ≤. That is why the circumference is 2πr and the area is πr², and the two results appear on separate cards.

Can I type the equation with x squared as x^2?

Yes. It accepts x^2, x² and products such as (x-2)^2. You can also paste the equation already in center form: the calculator expands it and extracts the center and radius.

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equation of a circle calculator

Equation of a circle: center form, general form, center and radius

Completing the square: from the general to the center form

The circle with center C(a, b) and radius r has the center-form equation (x - a)² + (y - b)² = r², which comes from the definition: every point is at distance r from the center. Expanding the squares gives the general equation x² + y² - 2ax - 2by + (a² + b² - r²) = 0, which the calculator also writes. For center (2, 3) and radius 5 the center form is (x - 2)² + (y - 3)² = 25 and the general form is x² + y² - 4x - 6y - 12 = 0, since 4 + 9 - 25 = -12. In the general form the x² and y² coefficients are equal and there is no xy term; without that the curve is not a circle (it may be an ellipse, for example).

To go from the general to the center form, group the x terms and the y terms and add to each group the square of half its coefficient, on both sides. In x² + y² - 4x + 6y - 12 = 0 this gives (x² - 4x + 4) + (y² + 6y + 9) = 12 + 4 + 9, that is, (x - 2)² + (y + 3)² = 25. The center is (2, -3), with the signs flipped relative to the parentheses, and the radius is √25 = 5. If the right-hand side comes out zero, the equation describes only the center point; if negative, no real circle exists, and the calculator tells you which case occurred instead of showing the root of a negative number.

There are other ways to define a circle. By the endpoints of a diameter, the center is the midpoint and r = AB/2: A(1, 2) and B(5, 4) give center (3, 3) and r² = 5, so r = √5. By three non-collinear points, you solve the perpendicular bisectors of two pairs: (0, 0), (4, 0) and (0, 2) give center (2, 1) and (x - 2)² + (y - 1)² = 5. The disk area is πr² and the circumference is 2πr, written exactly, such as 25π and 10π. With a point P entered, the squared distance to the center is compared with r²: smaller is inside, equal is on the curve, larger is outside.

Checked examples

Center and radius

Input: C(2, 3), r = 5
Expected output: (x - 2)² + (y - 3)² = 25; x² + y² - 4x - 6y - 12 = 0

Circumference 10π, area 25π.

General to center form

Input: x² + y² - 4x + 6y - 12 = 0
Expected output: (x - 2)² + (y + 3)² = 25; C(2, -3), r = 5

Added 4 + 9 to both sides.

Another negative center

Input: x² + y² + 2x - 4y - 4 = 0
Expected output: (x + 1)² + (y - 2)² = 9; C(-1, 2), r = 3

4 + 1 + 4 = 9.

From a diameter

Input: A(1, 2), B(5, 4)
Expected output: (x - 3)² + (y - 3)² = 5; r = √5

Circumference 2π√5.

Through three points

Input: (0, 0), (4, 0), (0, 2)
Expected output: (x - 2)² + (y - 1)² = 5

Center (2, 1).

No real circle

Input: x² + y² - 6x - 8y + 30 = 0
Expected output: r² = -5

Negative squared radius.

Full tool FAQ

How do you calculate the distance between two points?

How do you find the equation of the line through two points?

What is the difference between the standard, slope-intercept, intercept and parametric forms?

How do you calculate the slope and the angle of inclination?

How can you tell whether two lines are parallel or perpendicular?

How do you find the intersection of two lines?

How do you calculate the distance from a point to a line?

How do you check whether three points are collinear and find the triangle area?

How do you get the equation of a circle and go from general form back to center form?

Are the results exact? What are the limits?

Frequently asked questions

How do I find the center and radius from the general equation?

How do I know whether a line is secant, tangent or outside?

Are the circle and the disk the same thing?

Can I type the equation with x squared as x^2?

Yes. It accepts x^2, x² and products such as (x-2)^2. You can also paste the equation already in center form: the calculator expands it and extracts the center and radius.