A circle is a simple closed shape in Euclidean geometry. 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.

A circle is a simple closed curve which 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.

Parts of a CircleRADIUS

The radius of the circle is a straight line drawn from the center to the boundary line or the circumference. The plural of the word radius is radii. 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 the length of any of them. The name comes from 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 mathematic variable name for "radius" is r. By extension, the diameter d is defined as twice the radius:[3]

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.

The radius of the circle with perimeter (circumference) C is

Alternatively, this can be expressed as

DIAMETERIn geometry, a diameter of a circle is any straight line segment

that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere. The word "diameter" is derived from Greek διάμετρος (diametros), "diameter of a circle", from -δια (dia-), "across, through" + μέτρον (metron), "measure".[1] It is often abbreviated DIA, dia, d, or r.

In more modern usage, the length of a diameter is also called the diameter. In this sense one speaks of the diameter rather than a diameter (which refers to the line itself), because all diameters of a circle or sphere have the same length, this being twice the radius r.

For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the width is defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers.[2] For a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all such pairs of parallel tangent lines have the same distance.

For an ellipse, the standard terminology is different. A diameter of an ellipse is any chord passing through the midpoint of the ellipse.[3] For example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one. The longest diameter is called the major axis.

CIRCUMFERENCEThe circumference (from Latin circumferentia, meaning

"carrying around") of a closed curve or circular object is the linear distance around its edge.[1] The circumference of a circle is of special importance in geometry and trigonometry. Informally "circumference" may also refer to the edge itself rather than to the length of the edge. Circumference is a special case of perimeter: the perimeter is the length around any closed figure, but conventionally "perimeter" is typically used in reference to a polygon while "circumference" typically refers to a continuously differentiable curve.

The circumference of a circle is the distance around it. The term is used when measuring physical objects, as well as when considering abstract geometric forms.

CHORD

A chord of a circle is a straight line segment whose endpoints both lie on the circle. A secant line, or just secant, is the infinite line extension of a chord. More generally, a chord is a line segment joining two points on any curve, for instance an ellipse. A chord that passes through a circle's center point is the circle's diameter.The word chord is from the Latin chorda meaning bowstring.Chords of a circleAmong properties of chords of a circle are the follo wing:

1. Chords are equidistant from the center if and only if their lengths are equal.2. A chord that passes through the center of a circle is called a diameter, and is the longest chord.3. If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths

satisfy AP·PB = CP·PD (power of a point theorem).The area that a circular chord "cuts off" is called a circular segment.

ARCIn Euclidean geometry, an arc (symbol: ⌒) is a closed segment of

a differentiable curve. A common example in the plane (a two-dimensional manifold), is a segment of a circle called a circular arc.[1] In space, if the arc is part of a great circle (or great ellipse), it is called a great arc.

Every pair of distinct points on a circle determines two arcs. If the two points are not directly opposite each other, one of these arcs, the minor arc, will subtend an angle at the centre of the circle that is less than π radians (180 degrees), and the other arc, the major arc, will subtend an angle greater than radians.π

An arc is a part of the circumference between two points or a continuous piece of a circle. The shorter arc between P and Q is called the minor arc PQ. The longer arc between P and Q is called the major arc PQ.

SEMI-CIRCLEIn mathematics (and more specifically geometry), a

semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, π radians, or a half-turn). It has only one line of symmetry (reflection symmetry). In non-technical usage, the term "semicircle" is sometimes used to refer to a half-disk, which is a two-dimensional geometric shape that also includes the diameter segment from one end of the arc to the other as well as all the interior points.

By Thales' theorem, any triangle inscribed in a semicircle with a vertex at each of the endpoints of the semicircle and the third vertex elsewhere on the semicircle is a right triangle, with right angle at the third vertex.

All lines intersecting the semicircle perpendicularly are concurrent at the center of the circle containing the given semicircle.

Uses

A semicircle can be used to construct the arithmetic and geometric means of two lengths using straight-edge and compass. If we make a semicircle with a diameter of a+b, then the length of its radius is the arithmetic mean of a and b (since the radius is half of the diameter). The geometric mean can be found by dividing the diameter into two segments of lengths a and b, and then connecting their common endpoint to the semicircle with a segment perpendicular to the diameter. The length of the resulting segment is the geometric mean, [1] which can be proved using the Pythagorean theorem. This can be used to accomplish quadrature of a rectangle (since a square whose sides are equal to the geometric mean of the sides of a rectangle has the same area as the rectangle), and thus of any figure for which we can construct a rectangle of equal area, such as any polygon (but not a circle).

TANGENTIn geometry, the tangent line (or simply tangent) to a

plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve.[1] More precisely, a straight line is said to be a tangent of a curve y = f (x) at a point x = c on the curve if the line passes through the point (c, f (c)) on the curve and has slope f '(c) where f ' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.

Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.

The word "tangent" comes from the Latin tangere, 'to touch'.