Circle Center Calculator

Find the center and radius of a circle passing through three given points using the perpendicular bisector method.

Enter Three Points on the Circle

Point A
Point B
Point C

Result

Center of Circle
(3, 1.75)
coordinates (h, k)
Center x (h)3
Center y (k)1.75
Radius (r)2.25
Diameter (d)4.5
Circle Equation(x-3)²+(y-1.75)²=5.0625
Area15.904
Circumference14.137

Step-by-Step Solution

(x-3)² + (y-1.75)² = 5.0625

Finding the Center of a Circle from Three Points

Given three non-collinear points on the circumference of a circle, the center can be uniquely determined. This is because exactly one circle passes through any three non-collinear points in a plane. The method used is based on the perpendicular bisectors of the chords connecting these points.

The center of the circle is equidistant from all three points. This means it lies on the perpendicular bisector of every chord of the circle. By finding the intersection of two perpendicular bisectors, we locate the exact center.

The Perpendicular Bisector Method

Step 1: Find Midpoints

Calculate the midpoint of segment AB and segment BC.

M = ((x1+x2)/2, (y1+y2)/2)

Step 2: Find Slopes

Calculate the slope of each segment, then the negative reciprocal for the perpendicular bisector.

m_perp = -1 / m_chord

Step 3: Write Equations

Write the equation of each perpendicular bisector using point-slope form.

y - y_m = m_perp(x - x_m)

Step 4: Solve System

Solve the two perpendicular bisector equations simultaneously to find the center (h, k).

Center = intersection point

Step 5: Find Radius

Calculate the distance from the center to any of the three points.

r = sqrt((h-x1)² + (k-y1)²)

Step 6: Write Equation

Write the standard form circle equation.

(x-h)² + (y-k)² = r²

Alternative: Using the General Equation

Another approach is to substitute each point into the general circle equation x² + y² + Dx + Ey + F = 0. This gives three linear equations in D, E, and F, which can be solved simultaneously. The center is then (-D/2, -E/2) and the radius is r = sqrt(D²/4 + E²/4 - F).

Why Three Points?

Two points define infinitely many circles (any circle passing through both points). Three non-collinear points define exactly one circle. If the three points are collinear (lie on a straight line), no circle passes through all three -- the "circle" would have infinite radius, which is a straight line.

Practical Applications

Finding the center of a circle from three points has many practical uses in surveying, computer vision, archaeology, and engineering. When you can measure three points on a circular structure but cannot directly measure the center, this method allows precise determination of the center and radius.

  • Surveying: Determining the center and radius of a circular plot or curved boundary from field measurements.
  • Archaeology: Reconstructing circular structures (columns, wells, foundations) from fragmentary remains.
  • Computer vision: Fitting circles to detected edge points in image processing.
  • Engineering: Reverse-engineering circular components when original specifications are unavailable.
  • Navigation: Determining turning radius from GPS waypoints on a curved path.

Tips for Using This Calculator

  • The three points must not be collinear (on the same line).
  • Any three distinct non-collinear points uniquely define a circle.
  • Decimal and negative coordinates are fully supported.
  • The calculator also outputs the circle equation, area, and circumference.