Equation of a Circle Calculator
Our free circle calculator solves equation acircle problems. Get worked examples, visual aids, and downloadable results.
Formula
(x - h)^2 + (y - k)^2 = r^2
The standard form equation defines all points (x,y) at distance r from center (h,k). The general form x^2 + y^2 + Dx + Ey + F = 0 relates to the standard form by D = -2h, E = -2k, F = h^2 + k^2 - r^2.
Worked Examples
Example 1: Standard Form from Center and Radius
Problem: Write the equation of a circle with center (3, -2) and radius 5.
Solution: Center (h, k) = (3, -2), radius r = 5\nStandard form: (x - 3)^2 + (y + 2)^2 = 25\nGeneral form: x^2 + y^2 - 6x + 4y - 12 = 0\nD = -6, E = 4, F = -12\nCircumference = 2pi(5) = 31.4159\nArea = pi(25) = 78.5398
Result: (x - 3)^2 + (y + 2)^2 = 25 | Area = 78.540 sq units
Example 2: Convert General Form to Standard Form
Problem: Convert x^2 + y^2 - 6x + 4y - 12 = 0 to standard form.
Solution: D = -6, E = 4, F = -12\nCenter: h = -D/2 = 3, k = -E/2 = -2\nr^2 = h^2 + k^2 - F = 9 + 4 + 12 = 25\nr = 5\nStandard form: (x - 3)^2 + (y + 2)^2 = 25
Result: Center: (3, -2) | Radius: 5 | Standard: (x-3)^2 + (y+2)^2 = 25
Frequently Asked Questions
What is the standard form equation of a circle?
The standard form of a circle equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. This form directly reveals the center and radius, making it the most intuitive representation. The equation states that every point (x, y) on the circle is exactly r units away from the center (h, k), which is the geometric definition of a circle. For a circle centered at the origin, this simplifies to x^2 + y^2 = r^2. To graph a circle from standard form, plot the center point, then measure r units in all directions to find points on the circle. The standard form is particularly useful for identifying transformations: h represents a horizontal shift and k represents a vertical shift from the origin.
How do you find the equation of a circle from three points?
To find a circle through three non-collinear points (x1,y1), (x2,y2), (x3,y3), substitute each point into the general form x^2 + y^2 + Dx + Ey + F = 0 to get three linear equations in D, E, and F. For example, with points (1,1), (5,1), and (3,5): substituting gives 2 + D + E + F = 0, 26 + 5D + E + F = 0, and 34 + 3D + 5E + F = 0. Solving this system yields D, E, F, from which the center is (-D/2, -E/2) and radius is sqrt(D^2/4 + E^2/4 - F). An alternative geometric method finds the perpendicular bisectors of any two chords formed by the three points; their intersection is the center. Three collinear points do not define a circle. This construction appears in computational geometry and circumscribed circle calculations for triangles.
How do you determine if a point is inside, on, or outside a circle?
To determine a point's position relative to a circle, compute the distance from the point to the center and compare it to the radius. For point (px, py) and circle with center (h, k) and radius r, calculate d^2 = (px - h)^2 + (py - k)^2. If d^2 < r^2, the point is inside the circle. If d^2 = r^2, the point is exactly on the circle. If d^2 > r^2, the point is outside the circle. Using d^2 instead of d avoids computing a square root, which is a common optimization in computer graphics and game programming. Equivalently, substitute the point into the left side of the standard form equation: if the result is less than r^2, the point is inside. This test is fundamental to collision detection algorithms and geographic information systems.
What is the equation of a tangent line to a circle?
At a point (x1, y1) on a circle centered at origin with radius r, the tangent line equation is x1*x + y1*y = r^2. For a general circle (x-h)^2 + (y-k)^2 = r^2 with tangent point (x1, y1), the equation is (x1-h)(x-h) + (y1-k)(y-k) = r^2. The tangent line is perpendicular to the radius at the point of tangency. The slope of the radius from center (h,k) to point (x1,y1) is (y1-k)/(x1-h), so the tangent slope is -(x1-h)/(y1-k). For an external point, there are two tangent lines whose lengths equal sqrt((px-h)^2 + (py-k)^2 - r^2). Finding tangent lines is essential in optics for reflection calculations, in mechanical engineering for cam design, and in computer graphics for smooth curve rendering.
How do you find the intersection of a line and a circle?
To find where line y = mx + b intersects circle (x-h)^2 + (y-k)^2 = r^2, substitute the line equation into the circle equation: (x-h)^2 + (mx+b-k)^2 = r^2. Expanding gives a quadratic in x: (1+m^2)x^2 + 2(m(b-k)-h)x + (h^2+(b-k)^2-r^2) = 0. The discriminant determines the number of intersections: positive means two intersection points (secant line), zero means exactly one point (tangent line), and negative means no intersection (line misses the circle). For a vertical line x = c, substitute directly to get (y-k)^2 = r^2 - (c-h)^2. This calculation is fundamental to ray tracing in computer graphics, circle packing problems, and geometric construction algorithms.
What are the x-intercepts and y-intercepts of a circle?
The x-intercepts occur where y = 0, found by solving (x-h)^2 + k^2 = r^2, giving x = h plus or minus sqrt(r^2 - k^2). Real x-intercepts exist only when |k| is less than or equal to r (the circle reaches or crosses the x-axis). Similarly, y-intercepts occur where x = 0, found by solving h^2 + (y-k)^2 = r^2, giving y = k plus or minus sqrt(r^2 - h^2). Real y-intercepts exist when |h| is less than or equal to r. A circle can have 0, 1, or 2 intercepts on each axis. When |k| = r, the circle is tangent to the x-axis (one x-intercept). When the circle passes through the origin, both x = 0 and y = 0 satisfy the equation simultaneously. Intercepts are useful for graphing and for finding where circular paths cross reference axes.