How to Find the Centre and Radius of a Circle From Its Equation

Every point on a circle is the same distance from its centre. Writing that single fact in coordinates turns a circle into an equation, which can then tell you where the circle cuts the axes, whether two circles touch, and which circle passes through three given points.

This lesson covers the forms of a circle's equation, finding the centre and radius, circles through three points, and intercepts and relative positions.

**A circle with centre $(h,k)$ and radius r has standard form $(x-h{)}^2+(y-k{)}^2=r^2$, the circle on the diameter joining $(x_1,y_1)$ and $(x_2,y_2)$ is $(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$, expanding gives the general form $x^2+y^2+2gx+2fy+c=0$, and the parametric form is $x=h+r\cos \theta$, $y=k+r\sin \theta$.

Why the diameter form works. The angle in a semicircle is a right angle, so for any point on the circle the lines to the two ends of the diameter are perpendicular.

Worked example.** The circle with centre $(2,-3)$ and radius 5 is

$$(x-2{)}^2+(y+3{)}^2=25\Rightarrow x^2+y^2-4x+6y-12=0$$

Parametric form. For $x^2+y^2=16$, the points are $(4\cos \theta ,4\sin \theta )$; at $\theta =60°$ the point is $(2,2\sqrt{3})$.

An everyday example. Marking a circle for a rangoli with a string pinned at the centre uses the standard form — every chalk mark is one fixed length from the pin.

The substance. **The general form needs equal coefficients of $x^2$ and $y^2$ and no xy term** — without both, the equation is not a circle.

**For $x^2+y^2+2gx+2fy+c=0$, the centre is $(-g,-f)$ and the radius is $\sqrt{g^2+f^2-c}$, and the equation represents a real circle only when $g^2+f^2-c>0$.

Steps:**

- Divide so that the coefficients of $x^2$ and $y^2$ are 1
- Read $2g$ and $2f$ from the x and y terms
- Centre $(-g,-f)$ and radius $\sqrt{g^2+f^2-c}$

Worked example. Find the centre and radius of $2x^2+2y^2-8x+12y-6=0$.

- Divide by 2: $x^2+y^2-4x+6y-3=0$
- $2g=-4$ and $2f=6$, so $g=-2$, $f=3$ and $c=-3$
- Centre $(2,-3)$ and radius $\sqrt{4+9+3}=4$

An everyday example. Reading off the centre and reach of a circular sprinkler zone from a site plan's equation is exactly this conversion from general to standard form.

The substance. **Dividing by the coefficient of $x^2$ comes first** — $2x^2+2y^2-8x=0$ has centre $(2,0)$, not $(4,0)$.

**Substitute each of the three points into $x^2+y^2+2gx+2fy+c=0$ to get three linear equations in g, f and c, then solve them together; other data, such as the centre lying on a given line, is handled the same way, with one equation per condition.

Worked example.** Find the circle through $(0,0)$, $(4,0)$ and $(0,6)$.

- $(0,0)$: $c=0$
- $(4,0)$: $16+8g=0$, so $g=-2$
- $(0,6)$: $36+12f=0$, so $f=-3$

$$x^2+y^2-4x-6y=0$$

The centre is $(2,3)$ and the radius is $\sqrt{4+9}=\sqrt{13}\approx 3.61$.

An everyday example. Finding the centre of a round village well from three points marked on its rim is exactly this calculation.

The substance. Three collinear points give no circle — the three equations then have no solution.

**The circle $x^2+y^2+2gx+2fy+c=0$ cuts off $2\sqrt{g^2-c}$ on the x-axis and $2\sqrt{f^2-c}$ on the y-axis, and two circles whose centres are d apart, with radii $r_1$ and $r_2$, lie apart, touch, intersect or nest according to how d compares with $r_1+r_2$ and $|r_1-r_2|$.

Intercepts.** Putting $y=0$ gives $x^2+2gx+c=0$, whose roots differ by $2\sqrt{g^2-c}$.

Relative positions:

- $d>r_1+r_2$ — apart, outside each other
- $d=r_1+r_2$ — touching externally
- $|r_1-r_2|<d<r_1+r_2$ — intersecting at two points
- $d=|r_1-r_2|$ — touching internally
- $d<|r_1-r_2|$ — one inside the other

Worked example. For $x^2+y^2-6x-8y+9=0$: $g=-3$, $f=-4$ and $c=9$. The x-intercept is $2\sqrt{9-9}=0$, so the circle touches the x-axis; the y-intercept is $2\sqrt{16-9}=2\sqrt{7}\approx 5.29$.

Worked example 2. Circles with centres $(0,0)$ and $(6,8)$ and radii 4 and 6 have $d=10=4+6$, so they touch externally.

An everyday example. Two circular ripples spreading on a pond first touch externally and then cross at two points as they grow.

The substance. A zero intercept means tangency — the circle just touches that axis.