Find the equation of a circle from two endpoints of a diameter. Computes center (midpoint), radius, standard form, general form, area, and circumference.
Given two points that are endpoints of a diameter, you can completely determine the circle they define. The center of the circle is the midpoint of the diameter, and the radius is half the distance between the two points. From the center and radius, both the standard form and general form of the circle's equation follow immediately.
The midpoint formula, (h, k) = ((x₁+x₂)/2, (y₁+y₂)/2), gives the center. The distance formula, d = √((x₂−x₁)² + (y₂−y₁)²), gives the diameter, and dividing by 2 gives the radius r. The standard form equation is (x−h)² + (y−k)² = r², which expands into the general form x² + y² + Dx + Ey + F = 0 where D = −2h, E = −2k, and F = h² + k² − r².
This is a foundational problem in analytic geometry that appears throughout high school and college mathematics. It combines the midpoint formula, distance formula, and circle equations into one cohesive problem. The technique extends naturally to spheres in 3D.
This calculator takes two diameter endpoint coordinates (x₁, y₁) and (x₂, y₂), computes the center, radius, both equation forms, area, circumference, and the slope of the diameter, with a step-by-step summary table and presets for common configurations.
The Equation of a Circle from Diameter Endpoints Calculator is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Standard Form, General Form, Center (Midpoint) in one pass, with conversions and derived values shown together.
Use it to validate homework steps, check CAD or fabrication dimensions, estimate material requirements, and sanity-check hand calculations before submitting work.
Center: h = (x₁+x₂)/2, k = (y₁+y₂)/2 Diameter: d = √((x₂−x₁)²+(y₂−y₁)²) Radius: r = d/2 Standard form: (x−h)² + (y−k)² = r² General form: x² + y² − 2hx − 2ky + (h²+k²−r²) = 0 Area: A = πr² Circumference: C = 2πr
Result: Center = (1, 1), radius = 5, equation: (x−1)²+(y−1)² = 25
Midpoint: h = (−3+5)/2 = 1, k = (4+(−2))/2 = 1. Diameter = √(8²+6²) = √100 = 10. Radius = 5. Standard form: (x−1)²+(y−1)² = 25. General form: x²+y²−2x−2y−23 = 0.
This calculator combines the core geometry formula with the input mode selected in the interface, then derives companion values shown in the output cards, comparison bars, and reference tables. Use it to cross-check both direct calculations and reverse-solving scenarios where one measurement is unknown.
Equation of a Circle from Diameter Endpoints Calculator calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.
Keep units consistent across every input before interpreting area, perimeter, angle, or volume outputs. For best results, measure carefully, round only at the final step, and compare at least one manual calculation with the calculator output when building confidence.
Find the midpoint of the two points to get the center (h, k). Find the distance between them to get the diameter, then halve it for the radius r. The equation is (x−h)²+(y−k)² = r².
Two arbitrary points on a circle are not enough to determine it uniquely — you need three non-collinear points, or two points plus the constraint that they are endpoints of a diameter. Use this as a practical reminder before finalizing the result.
(x−x₁)(x−x₂) + (y−y₁)(y−y₂) = 0, where (x₁,y₁) and (x₂,y₂) are the diameter endpoints. Any point (x,y) on the circle satisfies this because the angle inscribed in a semicircle is 90°.
Expand (x−h)² and (y−k)², combine like terms, and move r² to the left side: x²+y²−2hx−2ky+(h²+k²−r²) = 0. Keep this note short and outcome-focused for reuse.
The diameter is vertical. The center is still the midpoint, and the equation works the same way. The diameter slope is undefined (vertical line).
Yes. Two diametrically opposite points determine a sphere. The center is the midpoint, the radius is half the distance, and the equation is (x−h)²+(y−k)²+(z−l)² = r².