A circle is the set of all points in a plane at a fixed distance (the radius) from a fixed centre point. Simple to define yet endlessly rich, the circle has inspired mathematical inquiry across every civilisation — from ancient attempts to compute π, to the impossible dream of squaring the circle, to the modern equation that defines it in coordinate geometry.
Tie a piece of string to a peg in the ground, pull it taut, and walk around the peg while keeping the string tight. The path you trace out is a circle — the set of every single point that sits at exactly the same distance from a central point. That fixed distance is the radius; twice the radius is the diameter.
For over 2,000 years, mathematicians tried to construct, using only a compass and an unmarked straightedge, a square with exactly the same area as a given circle. It was finally proven in 1882 (by Ferdinand von Lindemann, via proving π is transcendental) that this is mathematically impossible — no such compass-and-straightedge construction can ever exist, closing one of history's oldest open mathematical questions.
In coordinate geometry (see the Euclidean Geometry entry), a circle centred at (h,k) with radius r consists of exactly the points (x,y) satisfying:
This equation follows directly from the Pythagorean theorem — it simply states that the distance from (x,y) to the centre (h,k) equals r.
An angle inscribed in a circle (formed by two chords meeting at a point on the circle) is always exactly half the central angle that subtends the same arc. A striking special case: any angle inscribed in a semicircle is always exactly a right angle (Thales' theorem, attributed to Thales of Miletus, c. 600 BCE) — one of the earliest recorded pieces of deductive geometric reasoning in history.
A line tangent to a circle touches it at exactly one point, and is always perfectly perpendicular to the radius drawn to that point. This simple but powerful fact underlies countless geometric constructions and proofs.
For a point P outside a circle, and any line through P intersecting the circle at two points A and B, the product PA×PB is always the same constant value, regardless of which particular line through P is chosen — a quantity called the power of the point P with respect to the circle. This elegant invariant underlies many classical results in circle geometry, including the radical axis construction.
A circle is technically a special case of a more general family of curves called conic sections (see that entry) — specifically, the case where the "flattening" of an ellipse is reduced all the way to zero.
Babylonian tablets (c. 1900–1600 BCE) used π≈3.125. The Rhind Papyrus (Egypt, c. 1650 BCE) implies π≈3.16. Archimedes (c. 250 BCE) rigorously bounded 223/71<π<22/7 using inscribed and circumscribed 96-sided polygons (see the Archimedes entry). Zu Chongzhi (China, 5th century CE) computed π accurate to 7 decimal places using a 24,576-sided polygon, a record that stood for roughly 900 years. Mādhava (India, c. 1400 CE) discovered infinite series methods yielding 11 correct decimal digits (see the Mādhava entry) — each of these traditions independently pushing the boundary of achievable precision using entirely different techniques.
As covered in the Irrational Number entry, Lambert (1761) proved π is irrational, and Lindemann (1882) proved the far stronger fact that π is transcendental — not the root of any polynomial with integer coefficients. This transcendence is precisely what makes squaring the circle by compass and straightedge provably impossible, since compass-and-straightedge constructions can only ever produce specific algebraic (non-transcendental) lengths.
How densely can identical circles be packed together in the plane without overlapping? The optimal arrangement (hexagonal packing, where each circle touches 6 neighbours) achieves a density of exactly π/√12 ≈ 90.69% coverage — a result rigorously proven by Axel Thue (1890, with the proof later refined for full rigour by László Fejes Tóth, 1940). The analogous 3-dimensional sphere-packing problem (the Kepler Conjecture — see the Hilbert's 23 Problems entry) proved vastly harder, remaining unsolved until Thomas Hales's computer-assisted 1998 proof.
The familiar relationships C=2πr and A=πr² hold precisely only in flat, Euclidean space. In hyperbolic geometry, a circle's circumference grows exponentially with its radius rather than linearly; in spherical (elliptic) geometry, a "circle" of sufficiently large radius eventually shrinks back down to a single point at the sphere's antipode. This dependence of even the most basic circle formulas on the underlying geometry's curvature is a striking, concrete illustration of just how fundamentally the shift to non-Euclidean geometry (see the Euclidean Geometry entry) changes even the most elementary geometric facts.