An equation of a circle calculator writes the equation of a circle from its center and radius, or goes the other way and finds the center and radius from the general form. It also gives the diameter, circumference and area.
The circle equation formula comes in two forms: the standard form, which shows the center and radius directly, and the general form, which has the squared terms expanded.
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How the circle equation formula works
Standard form
A circle with center (h, k) and radius r has the equation:
(x − h)² + (y − k)² = r²
Every point on the circle is exactly r units from the center, which is the distance formula squared.
General form
Expanding the standard form gives the general form, where D = −2h, E = −2k and F = h² + k² − r².
x² + y² + Dx + Ey + F = 0
Going from general to standard form
To find the center and radius from the general form, use h = −D ÷ 2, k = −E ÷ 2 and r² = h² + k² − F. This is the same as completing the square. If r² is zero or negative, the equation is not a circle.
| Form | Equation | What it shows |
|---|---|---|
| Standard | (x − h)² + (y − k)² = r² | Center and radius |
| General | x² + y² + Dx + Ey + F = 0 | Expanded coefficients |
Worked examples
Example 1: center and radius
A circle has center (3, −2) and radius 5.
(x − 3)² + (y + 2)² = 25
Expanding gives x² + y² − 6x + 4y − 12 = 0, because 9 + 4 − 25 = −12. The area is 25π ≈ 78.54 and the circumference is 10π ≈ 31.42.
Example 2: from general form
Find the circle x² + y² + 8x − 6y + 9 = 0. Then h = −8 ÷ 2 = −4 and k = −(−6) ÷ 2 = 3, and r² = 16 + 9 − 9 = 16, so r = 4.
(x + 4)² + (y − 3)² = 16
Example 3: a circle at the origin
With center (0, 0) and radius 1 the equation is x² + y² = 1, the unit circle.
Frequently asked questions
What is the equation of a circle?
(x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius.
How do you find the center and radius from a general equation?
Complete the square, or use h = −D ÷ 2, k = −E ÷ 2 and r² = h² + k² − F.
How do you write the equation from a center and a point on the circle?
Use the distance from the center to the point as the radius. Find r with the distance formula, then put h, k and r into the standard form.
When is an equation not a circle?
When h² + k² − F is zero or negative. A value of zero is a single point, and a negative value has no real points.
How do I get the area and circumference?
Area is π × r² and circumference is 2 × π × r. The calculator shows both.