Circles in the Coordinate Plane

What is a circle, algebraically? A circle is a one-line idea: the set of points at a fixed distance from a given centre. With the distance formula in hand, that one-line idea becomes a single, beautiful equation that captures every circle in the plane, from a point on a planetary orbit to the rim of a pizza box. The equation $(x - h)^2 + (y - k)^2 = r^2$ is one of the most-used formulas in all of mathematics, science, and engineering.

Building the equation

Fix a centre $C = (h, k)$ and a radius $r > 0$. A point $(x, y)$ lies on the circle exactly when its distance from $C$ equals $r$. By the distance formula:

$\sqrt{(x - h)^2 + (y - k)^2} \;=\; r.$

Squaring both sides (both are non-negative, so this is safe) gives the standard form:

$(x - h)^2 + (y - k)^2 \;=\; r^2.$

When the centre is the origin ($h = k = 0$), the equation collapses to the cleanest form: $x^2 + y^2 = r^2$. The unit circle is $x^2 + y^2 = 1$.

• Computer Graphics: Rasterising a circle uses the implicit equation $(x - h)^2 + (y - k)^2 = r^2$; Bresenham's circle algorithm computes which pixels satisfy this most closely, integer-only, without trigonometry.
• Astronomy: Orbits are approximately circular (or elliptical for higher eccentricity); fitting an orbital path to telescope observations begins by fitting a circle equation to the angular positions over time.
• Manufacturing: CNC mills cut circular pockets by following the equation $(x - h)^2 + (y - k)^2 = r^2$ parameterised in time; the standard-form parameters $(h, k, r)$ are the inputs the operator enters directly.

Drag the centre. Adjust the radius slider. Watch the equation update live as $(x - h)^2 + (y - k)^2 = r^2$. The radius line marks the distance from the centre to any point on the circle.

Reading off the centre and radius

Given $(x - 3)^2 + (y + 2)^2 = 25$, you can read the centre and radius directly: the $x$-shift gives $h = 3$ (note the minus sign in $(x - h)$; if you see $(y + 2)$, that is $(y - (-2))$ so $k = -2$), and $r^2 = 25$ means $r = 5$. Centre $(3, -2)$, radius $5$.

The general form, and completing the square

If you expand the standard form, you get the general form:

$x^2 + y^2 + Dx + Ey + F = 0,$

where $D = -2h$, $E = -2k$, and $F = h^2 + k^2 - r^2$. To recover the centre and radius, you must complete the square: group the $x$-terms and $y$-terms, add and subtract the right constants. For example,

$x^2 + y^2 - 6x + 4y - 12 = 0$

rearranges to $(x^2 - 6x) + (y^2 + 4y) = 12$. Adding $(6/2)^2 = 9$ to the first group and $(4/2)^2 = 4$ to the second (also to the right side to balance):

$(x - 3)^2 + (y + 2)^2 = 12 + 9 + 4 = 25.$

Centre $(3, -2)$, radius $5$. Completing the square is the algebraic inverse of expansion, and is your main computational tool throughout coordinate geometry.

Try it

• Predict first: what equation describes a circle of radius $1$ centred at the origin? Place the centre at the origin, set $r = 1$, and verify the readout shows $x^2 + y^2 = 1$.
• Before adjusting: what is the equation of a circle of radius $3$ centred at $(2, -1)$? Drag the centre and slider to verify the readout shows $(x - 2)^2 + (y + 1)^2 = 9$.
• Confirm $(5, -1)$ lies on $(x - 2)^2 + (y + 1)^2 = 9$: $(5-2)^2 + (-1+1)^2 = 9 + 0 = 9$. Yes.

A trap to watch for

The signs in the standard form fool many beginners. The equation is $(x - h)^2 + (y - k)^2 = r^2$, with minus signs in both groups. A circle centred at $(-3, 4)$ has equation $(x - (-3))^2 + (y - 4)^2 = r^2$, which becomes $(x + 3)^2 + (y - 4)^2 = r^2$. The plus in $(x + 3)$ corresponds to a negative centre coordinate. Always flip the sign when reading off $h$ and $k$. Also: $r^2 = 25$ means $r = 5$, not $25$, do not forget the square root.

What you now know

You can write down the equation of any circle from its centre and radius, convert between standard and general form using completing-the-square, and verify whether a given point lies on a circle. The next section asks a deeper question: which points on a circle have rational coordinates? The answer connects circles to Pythagorean triples and number theory.

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 8, §3, the equation of a circle in standard form.
• Stewart, J. (2015). Calculus, 8th ed. Cengage. Appendix B reviews circle equations as a setup for calculus.
• Coxeter, H. S. M. (1969). Introduction to Geometry, 2nd ed. Wiley. Chapter 1 covers circles via the synthetic axioms.
• Apostol, T. M. (1967). Calculus, Volume 1, 2nd ed. Wiley. §1.5 develops circle equations and the parametrisation $(\cos\theta, \sin\theta)$.