Circle Equation from Three Points

The Circle from Three Points calculator computes center points and radius of a circle defined by three points not in a straight line.

INSTRUCTIONS: Enter the following:

• (P1) X and Y coordinates of Point 1
• (P1) X and Y coordinates of Point 2
• (P2) X and Y coordinates of Point 3

Circle from Three Points (CP): The calculator returns the center point and the radius as real number.

The Math / Science

The Circle from Three Points equation computes the center point and radius of a circle.  It does this by solving the general form of the equation of a circle (below) for the three coefficients (g, f and c). With g, f and c, one can then solve for the center points (h, k) and radius (r) as follows:

h = -1*g 
k = -1*f
`r = sqrt(h * h + k * k - c)`

The general form of the equation of a circle is:

x² + y² + 2gx + 2fy + c = 0

The common form of the equation of a circle, employing the radius and center points is:

r² = (x-h)² + (y-k)²

where:

• (h, k) is the center of the circle
• r is the radius

Circle Calculators

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• Area of Circle Sector f(r,Θ) - Computes the area of a sector (pie slice) of a circle given the radius (r) and angle (Θ).
• Angle of Circle Sector f(r,h) - Computes the angle in a circle from the radius and depth of the chord.
• Area of Circle Annulus - Computes the area of an annulus (ring) given the inner radius (r) and outer radius (R).
• Radius of Circle from Center and a Point - Computes the radius of a circle given the center point (h,k) and any other point (x,y) on the circle.
• Circumference of Circle -  Computes the circumference of a circle given the radius (C = 2 π r).
• Circle Arc Length - Computes the length of an arc length on a circle given the radius (r) and angle (Θ)
• Circle within a Triangle - Computes the radius of a circle inscribed within a triangle given the length of the three sides (a,b,c) of the triangle.
• Circle around a Triangle - Computes the radius of a circle that circumscribes a triangle given the length of the three sides (a,b,c) of the triangle.
• Circle Diameter from Area - Computes the radius and diameter of a circle from the area. 
• Circle Radius from Circumference - Computes the radius of a circle given the circumference.
• Circle Circumference from Area - Computes the circumference of a circle given the area.
• Circle Radius from Area - Computes the radius of a circle given the area.
• Chord Length: Computes the length of a chord in a circle from the radius and height.
• Chord Length from Arc Length and Radius: Computes the length of a chord on a circle based on the circle's radius (r) and the length of the arc (a).
• Circle Radius from Chord - Computes the radius of a circle based on the length of a chord and the chord's center height.
• Circle Equation from Center and one Point - Develops the general equation of a circle based on the coordinates of the center (h,k) and any point on the circle (x,y).
• Circle Equation from Three Points: Develops the general equation of a circle that goes through three points that are not in a straight line.
• Circle with same Perimeter as an Ellipse - Computes the radius of the circle with the same perimeter of an ellipse defined by the semi-major and semi-minor axes.
• Rectangles to Cover a Circle - Computes the number of rectangles needed to minimally cover a circle based on the circle's diameter and the length and width of the rectangles.