A circle is a simple closed shape. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius.
Video: Geometry: Introduction to Circles - radius, diameter, circumference and area of a circle
A circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc.
A circle may also be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.
Terminology
Annulus: the ring-shaped object, the region bounded by two concentric circles.
Arc: any connected part of the circle.
Centre: the point equidistant from the points on the circle.
Chord: a line segment whose endpoints lie on the circle.
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 which passes through the centre; or the length of such a line segment, which is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord, and it is twice the radius.
Disc: the region of the plane bounded by a circle.
Lens: the intersection of two discs.
Passant: a coplanar straight line that does not touch the circle.
Radius: a line segment joining the centre of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.
Sector: a region bounded by two radii and an arc lying between the radii.
Segment: a region, not containing the centre, bounded by a chord and an arc lying between the chord's endpoints.
Secant: an extended chord, a coplanar straight line cutting the circle at two points.
Semicircle: an arc that extends from one of a diameter's endpoints to the other. In non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, 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 touches the circle at a single point.
Chord, secant, tangent, radius, and diameter
Arc, sector, and segment
Analytic results
Length of circumference
Video: Circumference of a Circle
The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:
Circle with circumference C in black, diameter D in cyan, radius R in red, and centre or origin O in magenta.
In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the Latin radius, meaning ray but also the spoke of a chariot wheel.[1] The plural of radius can be either radii (from the Latin plural) or the conventional English plural radiuses.[2] The typical abbreviation and mathematical variable name for radius is r. By extension, the diameter d is defined as twice the radius:[3]
Video: Geometry: Introduction to Circles - radius, diameter, circumference and area of a circle
If an object does not have a center, the term may refer to its circumradius, the radius of its circumscribed circle or circumscribed sphere. In either case, the radius may be more than half the diameter, which is usually defined as the maximum distance between any two points of the figure. The inradius of a geometric figure is usually the radius of the largest circle or sphere contained in it. The inner radius of a ring, tube or other hollow object is the radius of its cavity.
For regular polygons, the radius is the same as its circumradius.[4] The inradius of a regular polygon is also called apothem. In graph theory, the radius of a graph is the minimum over all vertices u of the maximum distance from u to any other vertex of the graph.[5]
The radius of the circle with perimeter (circumference) C is
Formula
For many geometric figures, the radius has a well-defined relationship with other measures of the figure.
Circles
The radius of a circle with area A is
Video: How to Calculate the Radius of a Circle Given Its Circumference
ADVANCED READING MATERIAL (OPTIONAL)
The radius of the circle that passes through the three non-collinear points P1, P2, and P3 is given by
where θ is the angle ∠P1P2P3. This formula uses the law of sines. If the three points are given by their coordinates (x1,y1), (x2,y2), and (x3,y3), the radius can be expressed as
Video: How to find center of circle and radius (three non collinear points)
Regular polygons
A square, for example (n=4)
The radius r of a regular polygon with n sides of length s is given by r = Rns, where Values of Rn for .........
![{\displaystyle r={\frac {\sqrt {[(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}][(x_{2}-x_{3})^{2}+(y_{2}-y_{3})^{2}][(x_{3}-x_{1})^{2}+(y_{3}-y_{1})^{2}]}}{2|x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{1}y_{3}-x_{2}y_{1}-x_{3}y_{2}|}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/caffb954e8c39a26dcc5e4ad4f66494edd313008)
Values of Rn for .........
