Equation of a Circle Calculator
Generate the equation of a circle in both standard and general form. Enter the center coordinates (h, k) and the radius r to get both representations of the circle equation.
Standard Form of a Circle
The standard form equation (x - h)² + (y - k)² = r² is the most intuitive representation of a circle. The center point (h, k) tells you where the circle is positioned on the coordinate plane, and the radius r determines its size. Every point (x, y) that satisfies this equation lies exactly r units from the center.
This form is derived directly from the distance formula. The distance from any point on the circle to the center must equal r, so √((x-h)² + (y-k)²) = r. Squaring both sides gives the standard form. Students learning conic sections encounter this equation first because it clearly shows the geometric properties of the circle without any algebraic manipulation required.
Converting Between Forms
Converting from standard to general form involves expanding the squared terms and rearranging. Start with (x - h)² + (y - k)² = r², expand to get x² - 2hx + h² + y² - 2ky + k² = r², then move r² to the left side. The coefficients D = -2h, E = -2k, and F = h² + k² - r² give the general form.
Going the other direction requires completing the square. Group the x and y terms separately, complete the square for each, and identify h, k, and r from the result. This conversion skill is essential for analytic geometry problems where circles are given in general form and you need to graph them or find intersections with other curves.
Circles in Coordinate Geometry
Circle equations are fundamental in coordinate geometry and appear in problems involving tangent lines, chord lengths, intersections with other curves, and loci. A common problem type asks students to find the equation of a circle passing through three given points, which requires solving a system of three equations derived from substituting each point into the general form.
In analytic geometry, circles belong to the family of conic sections along with ellipses, parabolas, and hyperbolas. A circle is technically an ellipse with equal semi-axes. Understanding circle equations builds the foundation for studying these other conics. In applied fields, circles model orbits, gears, signal propagation patterns, and many other physical phenomena that involve constant-distance relationships.
Frequently Asked Questions
What is the standard form of a circle equation?
The standard form is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form makes it easy to identify the center and radius directly.
What is the general form?
The general form is x² + y² + Dx + Ey + F = 0. To convert from general to standard form, complete the square for both x and y terms.
How do you find the center from the general form?
From x² + y² + Dx + Ey + F = 0, the center is (-D/2, -E/2) and the radius is √(D²/4 + E²/4 - F).
What happens when the center is the origin?
When h = 0 and k = 0, the standard form simplifies to x² + y² = r², the simplest circle equation centered at the origin.
Can a circle equation have a negative radius?
No. The radius must be positive. If completing the square yields r² < 0, the equation has no real graph and represents an imaginary circle.