Enter Circle Construction Values
Provide two points on the circle and a radius. The tool then uses the perpendicular bisector of the chord to locate valid center points.
Example Data Table
This sample uses points A(1, 2), B(7, 6), and radius 5.
| Item | Example Value |
|---|---|
| Point A | (1.0000, 2.0000) |
| Point B | (7.0000, 6.0000) |
| Midpoint | (4.0000, 4.0000) |
| Chord length | 7.2111 units |
| Center 1 | (2.0785, 6.8823) |
| Center 2 | (5.9215, 1.1177) |
| Central angle | 92.2924° |
| Sagitta | 1.5359 units |
Formula Used
1) Midpoint of the chord
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
2) Chord length
d = √((x₂ - x₁)² + (y₂ - y₁)²)
3) Center distance from the midpoint
h = √(r² - (d/2)²)
4) Perpendicular unit direction
u⊥ = (-(y₂ - y₁)/d, (x₂ - x₁)/d)
5) Two possible circle centers
C₁ = M + h·u⊥ and C₂ = M - h·u⊥
6) Perpendicular bisector equation
(x₂ - x₁)(x - mₓ) + (y₂ - y₁)(y - mᵧ) = 0
7) Arc and sagitta
θ = 2 asin(d / 2r), arc = rθ, sagitta = r - √(r² - (d/2)²)
A real solution exists only when r ≥ d/2. If r = d/2, both center solutions collapse into one midpoint-centered circle.
How to Use This Calculator
- Enter the x and y coordinates for two distinct points on the circle.
- Enter a radius large enough to span both points.
- Choose which valid center you want highlighted in the summary.
- Select decimal precision and an optional unit label.
- Press Calculate Circle Geometry.
- Review the midpoint, bisector equation, both center locations, circle equations, and the Plotly graph.
- Use the CSV or PDF buttons to export the results section.
FAQs
1) What does this calculator find?
It finds the perpendicular bisector of a chord, the midpoint, both possible circle centers for a chosen radius, full circle equations, arc values, and a graph.
2) Why are there two centers?
A chord usually supports two circles of the same radius, one on each side of the chord. Their centers lie on the same perpendicular bisector.
3) When is there no real circle solution?
If the radius is smaller than half the chord length, the circle cannot pass through both points. The offset term becomes imaginary, so no real center exists.
4) What happens when the radius equals half the chord?
Then the chord is a diameter. The two center formulas return the same midpoint, so the solution collapses into one unique circle center.
5) Does the tool handle vertical and horizontal chords?
Yes. The formulas use vector geometry, so vertical chords, horizontal chords, and slanted chords all work correctly without special manual rearrangement.
6) What is the perpendicular bisector used for?
Any point on the perpendicular bisector is equally distant from the chord endpoints. That property makes it the natural path containing the possible circle centers.
7) Why show both standard and general circle equations?
The standard form makes the center and radius obvious. The general form is useful in algebra, graphing systems, coordinate proofs, and symbolic manipulation.
8) What does the Plotly graph help me verify?
It visually confirms the chord, midpoint, bisector direction, possible centers, and whether the chosen radius produces the expected circle geometry.