Circle Equation and Properties Tool
A circle is the set of all points in a plane whose distance from a fixed point (the
center) is a constant value (the radius).
This calculator converts between common circle equation forms and computes standard properties used in math,
engineering, design, and measurement.
1) Center–radius (standard) form
The most geometric form is:
\[
(x-h)^2 + (y-k)^2 = r^2
\]
where the center is \(C=(h,k)\) and the radius is \(r>0\).
The graph in this tool marks the center and a radius segment (a “compass opening”).
2) General form
Expanding the squares produces a quadratic equation in \(x\) and \(y\) with equal coefficients on \(x^2\) and \(y^2\).
A convenient general form is:
\[
x^2 + y^2 + Dx + Ey + F = 0
\]
Some sources also write \(A(x^2+y^2)+Dx+Ey+F=0\) with \(A\neq 0\). Dividing by \(A\) reduces it to the normalized
form above.
3) Convert general form → center & radius (complete the square)
Start from:
\[
x^2 + y^2 + Dx + Ey + F = 0
\]
Group \(x\)-terms and \(y\)-terms and complete the square:
\[
\left(x+\frac{D}{2}\right)^2 - \left(\frac{D}{2}\right)^2
+ \left(y+\frac{E}{2}\right)^2 - \left(\frac{E}{2}\right)^2
+ F = 0
\]
Move constants to the right:
\[
\left(x+\frac{D}{2}\right)^2 + \left(y+\frac{E}{2}\right)^2
=
\left(\frac{D}{2}\right)^2 + \left(\frac{E}{2}\right)^2 - F
\]
So the circle parameters are:
\[
h=-\frac{D}{2},\qquad k=-\frac{E}{2},\qquad
r^2 = h^2 + k^2 - F
\]
A real circle requires \(r^2>0\). If \(r^2=0\), the “circle” is a single point (degenerate). If \(r^2<0\), there is no real circle.
4) Convert center–radius → general form
Expand and collect like terms:
\[
(x-h)^2 + (y-k)^2 = r^2
\;\Rightarrow\;
x^2 + y^2 - 2hx - 2ky + (h^2+k^2-r^2)=0
\]
Therefore, in \(x^2+y^2+Dx+Ey+F=0\):
\[
D=-2h,\qquad E=-2k,\qquad F=h^2+k^2-r^2
\]
5) Circle from three points
Three non-collinear points determine a unique circle. One algebraic method is to use:
\[
x^2 + y^2 + Dx + Ey + F = 0
\]
and plug in each point \((x_i,y_i)\), giving three equations. Subtracting the first equation from the other two
eliminates \(F\) and yields a \(2\times 2\) linear system for \(D\) and \(E\). Once \(D\) and \(E\) are known:
\[
h=-\frac{D}{2},\qquad k=-\frac{E}{2},\qquad
F=-(x_1^2+y_1^2+Dx_1+Ey_1),\qquad
r^2=h^2+k^2-F
\]
If the determinant of the \(2\times2\) system is \(0\), the points are collinear and no unique circle exists.
6) Circle properties
Once the radius \(r\) is known, the classic measurements are:
- Diameter: \(d=2r\)
- Circumference: \(C=2\pi r\)
- Area: \(A=\pi r^2\)
7) Worked example
Given center \(C=(2,3)\) and radius \(r=4\):
\[
(x-2)^2 + (y-3)^2 = 16
\]
Properties:
\[
d=8,\qquad C=2\pi(4)=8\pi\approx 25.133,\qquad
A=\pi(4^2)=16\pi\approx 50.265
\]
Tip: In practical drawing/design, a “compass” sets the radius \(r\). This tool mirrors that by marking the center and a radius segment on the diagram.
