What is a Circle and Circumference?
A circle is a shape consisting of all points in a plane that are a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius.
The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the circumference C is related to the radius r and diameter d by 2πr = πd.
Hence, the formula to find the Circumference of a circle is: 2πr
In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.
One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons. The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and the corresponding formula–that the area is half the perimeter times the radius–namely, A = 1/2 × 2πr × r, holds in the limit for a circle.
Although often referred to as the area of a circle in informal contexts, strictly speaking the term disk refers to the interior of the circle, while circle is reserved for the boundary only, which is a curve and covers no area itself. Therefore, the area of a disk is the more precise phrase for the area enclosed by a circle.
- Annulus: a ring-shaped object, the region bounded by two concentric circles.
- Arc: any connected part of a circle. Specifying two end points of an arc and a center allows for two arcs that together make up a full circle.
- Centre: the point equidistant from all points on the circle.
- Chord: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments.
- Circumference: the length of one circuit along the circle, or the distance around the circle.
- Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius.
- Disc: the region of the plane bounded by a circle.
- Lens: the region common to (the intersection of) two overlapping discs.
- Passant: a coplanar straight line that has no point in common with the circle.
- Sector: a region bounded by two radii of equal length with a common center and either of the two possible arcs, determined by this center and the endpoints of the radii.
- Radius: a line segment joining the centre of a circle with any single point on the circle itself; or the length of such a segment, which is half (the length of) a diameter.
- Secant: an extended chord, a coplanar straight line, intersecting a circle in two points.
- Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the center of the circle to which their arc belongs to.
- Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as center. In non-technical common usage it may mean the interior of the two dimensional region bounded by a diameter and one of its arcs, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one.
- Tangent: a coplanar straight line that has one single point in common with a circle ("touches the circle at this point").
- All of the specified regions may be considered as open, that is, not containing their boundaries, or as closed, including their respective boundaries.
- The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetric inequality.)
- The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T.
- All circles are similar:
- A circle's circumference and radius are proportional.
- The area enclosed and the square of its radius are proportional.
- The constants of proportionality are 2π and π, respectively.
- The circle that is centred at the origin with radius 1 is called the unit circle.
- Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.
- Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points.
- Chords are equidistant from the centre of a circle if and only if they are equal in length.
- The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are:
- • A perpendicular line from the centre of a circle bisects the chord.
- • The line segment through the centre bisecting a chord is perpendicular to the chord.
- If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
- If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.
- If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplementary.
- • For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
- An inscribed angle subtended by a diameter is a right angle (see Thales' theorem).
- The diameter is the longest chord of the circle.
- • Among all the circles with a chord AB in common, the circle with minimal radius is the one with diameter AB.
- If the intersection of any two chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then ab = cd.
- If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then a2 + b2 + c2 + d2 equals the square of the diameter.
- The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8r 2 – 4p 2 (where r is the circle's radius and p is the distance from the centre point to the point of intersection).
- The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.
- A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to the circle.
- A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle.
- Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.
- If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the centre as O, the angles ∠BOA and ∠BPA are supplementary.
- If AD is tangent to the circle at A and if AQ is a chord of the circle, then ∠DAQ = 1/2arc(AQ).
- The chord theorem states that if two chords, CD and EB, intersect at A, then AC × AD = AB × AE.
- If two secants, AE and AD, also cut the circle at B and C respectively, then AC × AD = AB × AE. (Corollary of the chord theorem.)
- A tangent can be considered a limiting case of a secant whose ends are coincident. If a tangent from an external point A meets the circle at F and a secant from the external point A meets the circle at C and D respectively, then AF2 = AC × AD. (Tangent-secant theorem.)
- The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side of the chord (Tangent Chord Angle).
- If the angle subtended by the chord at the centre is 90 degrees then ℓ = r √2, where ℓ is the length of the chord and r is the radius of the circle.
- If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements