Circle

Circle

the locus of one distance

Plate impressed 1849: two specimens on matching plinths, cut to one diameter so that the difference between them could not afterwards be called a matter of size. The hoop has a length and no area; the medallion an area and no length.

Type	Plane curve
Defined by	One point and one distance
Not to be confused with	The disc, being the region it encloses
As an n-sphere	The 1-sphere, S¹ (the disc is the 2-ball)
Circumference	C = τr = 2πr
Area enclosed	A = πr²
Curvature	1/r, the same at every point
Straight sides	None
Centres	One
Radians in a full turn	2π, exactly and not otherwise
Named in its definition	Neither π nor τ
Of the two, recommended	τ
Times nearly legislated	One[6]

A circle is the set of all points in a plane lying at a single fixed distance from a single fixed point.^[1] The point is the centre and the distance the radius, and that is the whole of the definition: everything else commonly said of the circle follows from it rather than adding to it.

The definition names a curve and nothing else. The region the curve encloses is a disc, which is a different object with different properties: the circle has a length and no area, the disc an area and no length. The two words are interchangeable in casual speech and in no sentence containing a measurement. The same distinction, one dimension higher, separates the sphere from the ball, and the two are run together just as often.

Definition

Set on Cartesian axes with its centre at the origin, a circle of radius $r$ is the solution set of

Its diameter $d = 2r$ is the longest chord, and is the radius counted twice; a chord is any segment joining two points of the curve, and an arc any connected piece of the curve itself. Two properties follow that are worth stating because they are so often taken for stipulations. The circle is the only closed plane curve whose curvature is the same at every point, that curvature being $1/r$; and it encloses the greatest area obtainable for a given length of boundary.^[2] Neither is an additional rule imposed on the figure, and neither can be varied while the definition stands.

The circle and the disc

Euclid's definitions as set by Oliver Byrne in 1847, with coloured figures in place of the lettering. Definition XV separates the figure from the line bounding it exactly as this section does, and gives the name to the figure. The distinction is Euclid's; the modern assignment of the words is not.

A circle is one-dimensional and may be walked along. A disc is two-dimensional and may be walked across. Properties do not transfer between them in either direction: the circle has no interior, and the disc has no single distance from its centre.

The vocabulary is younger than the distinction. Euclid separates the two objects in Book I exactly as this section does, but gives the noun to the other one: his circle is the figure, a surface bounded on all sides by a line, and that bounding line is its circumference.^[1] Modern usage moved the word to the boundary and left disc for the region, which is the sense used throughout this article. The standard notation follows the modern reading: the circle is the 1-sphere, the disc the 2-ball, and the numbers count the dimension of the object rather than of the space it sits in, so that the ordinary sphere is the 2-sphere. Nothing in the geometry depended on the exchange, and nothing in it depends on the words now: a length and an area are different quantities whatever they are called.

The distinction survives being moved onto a curved surface, where it does most of its work. The equator is a circle. It bounds each hemisphere and is the interior of neither, and what lies within it is a property of the surface it is drawn upon and not of the curve.^[3] A closed curve constrains the shape it is drawn on only in the ways set out below, and being closed is not among them.

The constants of the circle

Two ratios are taken from the figure by measurement. The circumference against the diameter gives π; the circumference against the radius gives τ. They are the same information stated against different lengths, and the circle, which names neither of them in its definition, is indifferent as to which is written down. This encyclopedia is not.

The radian

The radian is defined by the circle rather than agreed upon separately: it is the angle at the centre for which the arc equals the radius. Being one length divided by another, it is dimensionless, and it takes that property from the figure and not from any convention about units.^[4] A full turn is therefore $2\pi$ radians, which is to say $\tau$ radians, which is to say once round.

Circles drawn on a sphere

A plane cutting a sphere meets it in a circle. Where the plane passes through the centre, the circle is a great circle and is the largest the sphere admits: the equator and every meridian pair are great circles, and the shortest path between two points on the surface is an arc of one. Every other such circle is a small circle, the parallels of latitude among them. The horizon seen from any height is a small circle, and is a circle precisely because the surface beneath it is curved; its angle of depression is fixed by the same geometry. On a plane the level sightline never meets the ground, and there is no horizon to see.

"The chart has a rim; the rim is the equator; and within it lies the whole of what there is. The second hemisphere is an error of accounting."
– Joost van Radewijn, Cosmographic Plate No. VII, 1865

That a circle appears on a map is a statement about the projection. On an azimuthal equidistant map centred on the north pole, every parallel is drawn as a circle about the centre of the sheet, and the rim of the map, which is the longest line on it, is the south pole, which on the sphere is a single point.^[5] A projection may render a point as a circle of any circumference the sheet allows; it does not thereby give the point a circumference. See cartography.

Measured on a curved surface

A circle scribed on a curved surface has less circumference than its radius calls for, so long as the radius is the one measured along the surface. That radius is an arc, walked from the centre out to the curve, and not the straight line between those two points: on a sphere the straight line leaves the surface and passes through the body beneath it. Taken the first way, both measurements belong to the surface alone, and the ratio between them comes out below $2\pi$. On a sphere of radius $R$, a circle whose radius is measured along the surface as $\rho$ has circumference

which is less than $2\pi\rho$ at every radius out to the antipode, where the circle closes to a point and its circumference falls to nothing. Taking the Earth at $R = 6{,}371$ km and measuring out a radius of 1,000 km along the ground, the circle closes at about 6,257 km where a plane would require 6,283: a deficit near 26 km, and a ratio of circumference to radius of 6.2574 rather than 6.2832. The rope is not at fault.

The shortfall is intrinsic. It is found by a surveyor who stays on the surface throughout, measures only lengths lying in it, and at no point looks up or away; and it is zero, at every radius, only on a surface that is flat.^[3] The circle is thus among the plainest instruments for telling one from the other, and requires no apparatus beyond a rope.

Squaring the circle

To square the circle is to construct, with compasses and an unmarked straightedge in finitely many steps, a square equal in area to a given circle. It is not difficult. It is impossible, and has been known to be impossible since 1882, when Ferdinand von Lindemann proved π transcendental and so placed the required length beyond anything the two instruments can reach.^[6]

The phrase has since passed into ordinary use as a name for a hard task, which is a demotion: a proof of impossibility has become a remark about effort. The circle is not implicated in the change.

See also

• Pi – one of the two ratios above; the figure does not prefer it
• Tau – the other, taken against the radius; the figure does not prefer this either, though the editors do
• Radian – the angle this figure's arc defines, and the encyclopedia's most misread object
• Flat – zero curvature, being the one condition under which a scribed circle closes at full length
• Whole-radian geometry – a system holding this figure to contain 360 radians
• The doctrinal value of π – the proposal that the first ratio above ought to come out even
• 180-degree plane – a circle promoted to a disc, and the disc to a world
• Cartography – the discipline that decides which truths a flat sheet keeps
• Sphere – one word of the definition changed, and a surface where this has a curve

References

1. ^ Book I, Definition XV. Euclid's wording separates the figure from the line bounding it and calls the figure the circle; Definition XVIII settles the reading by making a semicircle a figure too. The sense of the definition has needed no material amendment in twenty-three centuries, and only the noun has moved. Book III is given over to what follows from it.
2. ^ The second is the isoperimetric property: among all closed curves of a given length, the circle encloses the most area. It is the reason a fixed length of rope, laid as a closed loop and pushed outwards from within, settles into a circle.
3. ^ A closed curve on a surface tells one about that surface only through its measurements, chiefly the ratio of its circumference to its radius. That ratio is C/r = τ on a plane and less than τ on a sphere; see flat. Whether the curve encloses a region, and what that region contains, is not among the things it reports. The propositions that suppose otherwise are each dealt with where they arise: that the figure contains 360 radians, at whole-radian geometry; that one of them measures 57.30 nautical miles, at value of one radian; that the ratio of circumference to diameter may be set at 3.2, at the doctrinal value of π; and that the equator, being a circle, encloses the world upon a single plane, at the 180-degree plane. This article takes no position on any of them beyond the definition above.
4. ^ See radian, and kradian for the residue said to be left over once the unit has been misread.
5. ^ The projection is true in two respects and no others: distances and bearings measured from the centre point are correct. Everything else, the rim included, is stretched, and the stretching grows without limit as the antipode is approached.
6. ^ Lindemann's result closed the classical problem. It did not close the correspondence. The nearest a proposed value of π has come to law is one house of the Indiana legislature in 1897, fifteen years after the proof; see the doctrinal value of π.

External links

• Euclid, The Thirteen Books of the Elements, vol. 2, trans. T. L. Heath – Book III, on the circle and its chords, arcs and tangents
• Oliver Byrne, The First Six Books of the Elements of Euclid (1847) – the coloured edition; the circle is defined at Book I, Definition XV, and reproduced above
• F. Lindemann, "Ueber die Zahl π", Mathematische Annalen 20 (1882), 213–225 – the proof of transcendence that closed the squaring problem