Circle — Standard Forms

Stand at the centre of a cricket pitch. Tie one end of a rope to the stumps, hold the other end, and walk until the rope is taut. Now walk in any direction you like — north, southeast, west-southwest — keeping the rope taut the whole time. The path your feet trace is a circle.

That is the geometric definition, and it is as old as civilisation. Indian mathematicians knew it well: Aryabhata, writing in the 5th century, computed \pi to four decimal places by studying exactly this shape. The Sulba Sutras, centuries earlier, gave constructions for circles inscribed in squares and squares inscribed in circles.

But the geometric definition — "all points equidistant from a centre" — doesn't, by itself, let you compute anything. You can draw a circle with a compass, but you can't easily answer questions like: Does the point (5, 1) lie inside, on, or outside this circle? Where does a given line cut through it? For that, you need an equation.

The equation of a circle is one of the cleanest objects in coordinate geometry. It comes directly from the definition, using nothing more than the distance formula, and it takes two forms — one built for understanding, one that falls out of exam problems. This article derives both from scratch and shows you how to move between them. By the end, you will be able to look at any circle equation — however messy it appears — and immediately read off the centre, the radius, and everything you need for the problems that come next.

From definition to equation

A circle has a centre C = (h, k) and a radius r > 0. A point P = (x, y) lies on the circle if and only if the distance from P to C equals r.

By the distance formula:

Square both sides to remove the square root:

Why: squaring is safe here because both sides are non-negative. No information is lost.

That is the entire derivation. The equation says exactly what the definition says: the squared distance from (x, y) to (h, k) equals r^2. Every point satisfying this equation is on the circle; every point that doesn't, isn't.

Special case: centre at the origin

When the centre is the origin (0, 0), the equation simplifies to

This is the simplest circle equation. A circle of radius 5 centred at the origin is x^2 + y^2 = 25. You can verify: the point (3, 4) lies on it because 9 + 16 = 25, and so does (0, 5) because 0 + 25 = 25, and so does (-5, 0) because 25 + 0 = 25.

A circle with centre away from the origin

Take a circle with centre (2, 3) and radius 4. Its equation is

The point (6, 3) lies on it: (6 - 2)^2 + (3 - 3)^2 = 16 + 0 = 16. So does (2, 7): (2 - 2)^2 + (7 - 3)^2 = 0 + 16 = 16. These are the points directly to the right of and directly above the centre, each at distance 4.

Reading the equation

The standard form is designed so that you can read the centre and radius at a glance:

• Centre: the numbers being subtracted from x and y inside the brackets. In (x - 2)^2 + (y - 3)^2 = 16, the centre is (2, 3).
• Radius: the square root of the number on the right side. Here, r = \sqrt{16} = 4.

A common slip: the equation (x + 1)^2 + (y - 5)^2 = 9. The centre is (-1, 5), not (1, 5). The sign flips because x + 1 = x - (-1).

The general form

The standard form is clean, but it is not what you usually encounter in problems. Expand (x - 2)^2 + (y - 3)^2 = 16:

Collect everything on the left:

This is the general form — it looks like a generic second-degree equation with no xy term and equal coefficients on x^2 and y^2 (both are 1).

The general pattern, for any circle, is:

The letters g, f, c are traditional — they come from the general second-degree equation ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0, which in the circle case has a = b = 1 and h = 0.

Why the centre is (-g, -f)

Start from the general form and complete the square to recover the standard form. This derivation is the key conversion tool.

Take x^2 + y^2 + 2gx + 2fy + c = 0.

Completing the square in x:

Why: to complete the square for x^2 + 2gx, take half the coefficient of x — that's g — and add and subtract g^2. This is the same geometric trick from the quadratic-formula derivation: the missing corner of the square is (g)^2.

Completing the square in y:

Substituting back:

This is now in standard form (x - h)^2 + (y - k)^2 = r^2, with:

• h = -g, so the centre's x-coordinate is -g
• k = -f, so the centre's y-coordinate is -f
• r^2 = g^2 + f^2 - c

The signs flip because (x + g) = (x - (-g)). And the expression g^2 + f^2 - c must be positive for the equation to represent a real circle with a positive radius. If g^2 + f^2 - c = 0, the "circle" degenerates to a single point (the centre). If g^2 + f^2 - c < 0, no real points satisfy the equation at all — it is an imaginary circle.

Matching the coefficients

When a problem gives you x^2 + y^2 - 4x - 6y - 3 = 0, compare with x^2 + y^2 + 2gx + 2fy + c = 0:

• 2g = -4, so g = -2
• 2f = -6, so f = -3
• c = -3

Centre = (-g, -f) = (2, 3) and r = \sqrt{g^2 + f^2 - c} = \sqrt{4 + 9 + 3} = \sqrt{16} = 4.

That recovers exactly the circle you started with: centre (2, 3), radius 4. The two forms describe the same circle, just wearing different clothes.

Worked examples

Common confusions

A few traps that catch students reliably.

• "The equation (x - 3)^2 + (y + 2)^2 = 16 has centre (3, 2)." The centre is (3, -2), not (3, 2). The y + 2 term means y - (-2), so k = -2. The sign inside the bracket is opposite to the coordinate of the centre. Always mentally rewrite + as -(-\text{something}).

• "r^2 = 16 so r = 16." No — r = 4. The right side of the standard form is r^2, not r. This mistake doubles your radius and quadruples the area.

• "Every equation of the form x^2 + y^2 + 2gx + 2fy + c = 0 is a circle." Only if g^2 + f^2 - c > 0. If g^2 + f^2 - c = 0, the equation represents a single point. If g^2 + f^2 - c < 0, there are no real solutions — the equation represents nothing you can draw.

• "I can just read off g and f directly." Be careful with signs. If the equation is x^2 + y^2 - 10x + 4y + 13 = 0, then 2g = -10 gives g = -5, and 2f = +4 gives f = 2. The centre is (-g, -f) = (5, -2). Every sign matters, and there are three sign-flips in play: from 2g to g, from g to -g, and from the bracket sign. Check by substituting the centre back into the equation.

• "The coefficient of x^2 can be anything." In the standard general form, the coefficients of x^2 and y^2 are both 1. If a problem gives you 2x^2 + 2y^2 + 8x - 12y + 6 = 0, divide through by 2 first: x^2 + y^2 + 4x - 6y + 3 = 0. Only then can you read off g, f, c.

Going deeper

When is a second-degree equation a circle?

The most general second-degree equation in x and y is

This equation represents a circle when three conditions are met:

1. a = b — the coefficients of x^2 and y^2 are equal.
2. h = 0 — there is no xy term.
3. g^2 + f^2 - ac > 0 — the radius is real and positive (after dividing by a to make the leading coefficient 1).

Conditions 1 and 2 together guarantee that the equation treats x and y symmetrically — which is exactly what a circle does (it looks the same in every direction from its centre). If a \neq b, the shape is stretched more in one direction than the other, giving an ellipse. If h \neq 0, the shape is rotated, which a circle (being perfectly symmetric) would never need.

The circle as a locus

In the language of locus, a circle is the locus of a point that moves so that its distance from a fixed point remains constant. The "fixed point" is the centre; the "constant" is the radius.

This phrasing is not just an alternative way of saying the same thing — it is a method. Many exam problems say "Find the locus of a point P such that..." and the answer turns out to be a circle. The way you solve such problems is:

1. Let P = (x, y).
2. Translate the geometric condition into an equation involving x and y.
3. Simplify. If the equation takes the form (x - h)^2 + (y - k)^2 = r^2, or equivalently x^2 + y^2 + 2gx + 2fy + c = 0 with g^2 + f^2 - c > 0, the locus is a circle.

For instance: find the locus of a point P such that PA^2 + PB^2 = k, where A = (1, 0) and B = (-1, 0) and k is a positive constant.

Let P = (x, y). Then:

Setting this equal to k:

This is a circle centred at the origin with radius \sqrt{(k - 2)/2}, provided k > 2. If k = 2, the locus is just the origin. If k < 2, no such point exists.

Concentric circles

Two circles are concentric if they share the same centre but have different radii. Their equations are (x - h)^2 + (y - k)^2 = r_1^2 and (x - h)^2 + (y - k)^2 = r_2^2. In general form, they differ only in the constant term c — the g and f are the same (since the centre is the same).

The equation of any circle concentric with x^2 + y^2 + 2gx + 2fy + c = 0 can be written as x^2 + y^2 + 2gx + 2fy + c' = 0 for some different constant c'. This is useful when a problem says "find the circle concentric with a given circle that passes through a given point" — you keep g and f, solve for c' using the point.

Historical note

The circle has been studied longer than almost any other curve. The ratio of a circle's circumference to its diameter — the number \pi — has fascinated mathematicians across every civilisation. Aryabhata (499 CE) gave \pi \approx 3.1416 and stated that it was "approximate" (asanna), recognising implicitly that \pi is irrational. Madhava of Sangamagrama (14th century) computed \pi to 11 decimal places using infinite series — work that predated similar European results by two centuries. Bhaskara II, in the 12th century, used properties of circles extensively in the Lilavati, including methods for computing chord lengths and arc lengths that anticipated coordinate-geometric ideas.

The equation of a circle in coordinate form, though, is a product of the coordinate-geometry revolution. Once Descartes and others established that curves could be described by equations, the circle — the simplest closed curve — was naturally the first to be written down. Its equation is the foundation on which all of conic-section geometry rests.

Where this leads next

You now know how to write the equation of a circle and convert between its two forms. The next articles build on this in several directions.

• Circle — Special Cases — the circle through three given points, the diameter form, parametric equations, and how to test whether a point is inside or outside a circle.
• Line and Circle — what happens when a straight line meets a circle: intersection, tangency, chord lengths, and the equation of a tangent.
• Tangent and Normal to Circle — the full theory of tangent lines, including the condition for tangency and the equation of the normal.
• Locus — the general method of describing curves as paths traced by moving points, of which the circle is the simplest example.
• Coordinate Geometry — Basics — the distance formula and section formula that this article relies on.