Formula and Method for the Equation of a Circle from Diameter Endpoints
Any two points in the plane can serve as the endpoints of a circle's diameter, and that alone is enough information to pin down the circle completely. Because a diameter is a straight line segment through the center, the midpoint of the two endpoints must be the circle's center, and half the length of that segment must be the radius. Once you know the center (h, k) and radius r, the circle's standard equation follows immediately: (x − h)² + (y − k)² = r².
How the calculation works
Given endpoints A(x₁, y₁) and B(x₂, y₂), the calculator finds the center by averaging the coordinates: h = (x₁ + x₂) / 2 and k = (y₁ + y₂) / 2. It then finds the diameter length with the distance formula, d = √((x₂ − x₁)² + (y₂ − y₁)²), and halves it to get the radius, r = d / 2. Substituting h, k, and r into (x − h)² + (y − k)² = r² gives the standard equation. There is also a direct route that skips computing the center and radius separately: by Thales' theorem, any point (x, y) on a circle sees a diameter at a right angle, so the vectors from (x, y) to each endpoint are perpendicular and their dot product is zero: (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0. Expanding either form gives the same general equation, x² + y² + Dx + Ey + F = 0, with D = −(x₁ + x₂), E = −(y₁ + y₂), and F = x₁x₂ + y₁y₂.
Common mistakes
- Using the full distance as the radius: the distance between the two endpoints is the diameter, not the radius — forgetting to divide by 2 doubles the circle's size.
- Sign errors in the standard form: if the center has a negative coordinate, the term flips sign — a center at h = −3 gives (x + 3)², not (x − 3)².
- Assuming the two points are the center and a point on the circle: that is a different problem (radius = distance between the points, not half of it) and gives a circle twice as large as the diameter-endpoints case.
- Averaging squares instead of coordinates: the center comes from averaging x-values and y-values separately, not from squaring or combining them any other way.
Real-world applications
- Coordinate geometry proofs and homework that ask you to derive a circle's equation from two given points.
- CAD, CNC, and vector-drawing software that define a circular arc or hole from two diametrically opposite reference points.
- Surveying and mapping tasks that need a circle (service radius, buffer zone) centered exactly between two known landmarks.
- Mechanical design, where a bolt-hole or bore is specified by two opposite points on its edge rather than a center and radius.