In classical geometry Geometry is a part of mathematics concerned with questions of size, shape, relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the third century BC geometry was put into an axiomatic form by Euclid, whose treatment—, a radius of a circle A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the centre. The common distance of the points of a circle from its center is called its radius or sphere A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The is any line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment is either an edge if they are adjacent vertices, or from its center to its perimeter A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter (measure). The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference. By extension, the radius of a circle or sphere is the length Length is the long dimension of any object. The length of a thing is the distance between its ends, its linear extent as measured from end to end. This may be distinguished from height, which is vertical extent, and width or breadth, which are the distance from side to side, measuring across the object at right angles to the length. In the of any such segment, which is half the diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle. The word "diameter" derives from Greek διάμετρος , "diagonal of a circle", from δια- (dia-), "across, through&.^[1]

More generally — in geometry Geometry is a part of mathematics concerned with questions of size, shape, relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the third century BC geometry was put into an axiomatic form by Euclid, whose treatment—, science Science is, in its broadest sense, any systematic knowledge-base or prescriptive practice that is capable of resulting in a prediction or predictable type of outcome. In this sense, science may refer to a highly skilled technique or practice, engineering Engineering is the discipline, art and profession of acquiring and applying technical, scientific and mathematical knowledge to design and implement materials, structures, machines, devices, systems, and processes that safely realize a desired objective or inventions, and many other contexts — the radius of something (e.g., a cylinder A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since, a polygon In geometry a polygon is traditionally a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners. The interior of the polygon is, a mechanical part, a hole, or a galaxy A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The name is from the Greek root galaxias [γαλαξίας], meaning "milky," a reference to the Milky Way galaxy) usually refers to the distance Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves from its center In geometry, the centre of an object is a point in some sense in the middle of the object. If geometry is regarded as the study of isometry groups then the centre is a fixed point of the isometries or axis of symmetry Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection to a point in the periphery: usually the point farthest from the center or axis (the outermost or maximum radius), or, sometimes, the closest point (the short or minimum radius).^[2] If the object does not have an obvious center, the term may refer to its circumradius, the radius of its circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the circumcenter or circumscribed sphere In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing. When it exists, a circumscribed sphere need not be the smallest sphere containing the polyhedron; for instance, the tetrahedron formed by a. 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.

The radius of a regular polygon (or polyhedron) is the distance from its center to any of its vertices; which is also its circumradius.^[3] The inradius of a regular polygon is also called apothegm An adage , or adagium (Latin), is a short but memorable saying that holds some important fact of experience that is considered true by many people, or that has gained some credibility through its long use. It often involves a planning failure such as "don't count your chickens before they hatch" or "don't burn bridges behind you.&.

In graph theory In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of vertices. A graph may be undirected, meaning, the radius of a graph In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path connecting them. This is also known as the geodesic distance because it is the length of the graph geodesic between those two vertices. If there is no path connecting the two vertices, i.e., if they belong to different is the minimum over all vertices u of the maximum distance from u to any other vertex of the graph.^[4]

The name comes from Latin Latin is an Italic language originally spoken in Latium and Ancient Rome. Through the Roman conquest, Latin spread throughout the Mediterranean and a large part of Europe. Romance languages such as Catalan, French, Italian, Portuguese, Romanian, and Spanish are descended from Latin, while many others, especially European languages, have inherited radius, meaning "ray" but also the spoke of a chariot wheel. The plural in English English is a West Germanic language that developed in England during the Anglo-Saxon era. As a result of the military, economic, scientific, political, and cultural influence of the British Empire during the 18th, 19th, and early 20th centuries, and of the United States since the mid 20th century, it has become the lingua franca in many parts of is radii (as in Latin), but radiuses can be used, though it rarely is.^[5]

Formulas for circles

Radius from circumference

The radius of the circle with perimeter A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter (measure). The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference (circumference where r is the radius and d is the diameter of the circle, and π is defined as the ratio of the circumference of the circle to its diameter (the numerical value of pi is 3.141 592 653 589 793...)) C is

Radius from area

The radius of a circle with area Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron. Area is an important invariant in the differential A is

Radius from three points

To compute the radius of a circle going through three points P₁, P₂, P₃, the following formula can be used:

where θ is the angle

Formulas for regular polygons

These formulas assume a regular polygon with n sides.

Radius from side

The radius can be computed from the side s by:

where

Formulas for hypercubes

Radius from side

The radius of a d-dimensional hypercube with side s is

See also

• Radius (bone) The radius is the bone of the forearm that extends from the lateral side of the elbow to the thumb side of the wrist. The radius is situated on the lateral side of the ulna, which exceeds it in length and size. It is a long bone, prism-shaped and slightly curved longitudinally. The radius articulates with the capitulum of the humerus
• Radius of curvature Radius of curvature is a term characterizing the measure of how curved, or bent, a given curve or surface is. Different disciplines have different conventions concerning its definition and use. The following articles deal with various aspects:
• Bend radius Bend radius, which is measured to the inside curvature, is the minimum radius one can bend a pipe, tube, sheet, cable or hose to without kinking it, damaging it, or shortening its life. The smaller the bend radius, the greater is the material flexibility . The diagram below illustrates a cable with a seven-centimeter bend radius
• Radius of convexity This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology
• Radius of convergence In mathematics, the radius of convergence of a power series is a non-negative quantity, either a real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well. If the series converges, it is the Taylor series of the analytic
• Radius of gyration Radius of gyration is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the root mean square distance of the objects' parts from either its center of gravity or an axis
• Atomic radius To a first approximation, atoms generally behave like minute spherical objects. The atomic radius of a chemical element is a measure of the size of its atoms, usually the distance from the nucleus to the boundary of the surrounding cloud of electrons. Since the boundary is not a well-defined physical entity, there are various non-equivalent
• Bohr radius In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central nucleus. The model says that the electrons orbit only at certain distances from the nucleus, depending on their energy. In the simplest atom, hydrogen, a single electron orbits, and the smallest possible orbit for the electron, that with the
• Filling radius In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating Systolic in Riemannian geometry
• Schwartzschild radius The Schwarzschild radius is a characteristic radius associated with every quantity of mass. It is the radius of a sphere in space, that if containing a correspondingly sufficient amount of mass (and therefore, reaches a certain density), the force of gravity from the contained mass would be so great that no known force or degeneracy pressure could

References

1. ^ Definition of radius at mathwords.com. Accessed on 2009-08-08.
2. ^ Robert Clarke James, Glenn James (1992), Mathematics dictionary. 548 pages, Springer ISBN 0412990415, 9780412990410
3. ^ Barnett Rich, Christopher Thomas (2008), Schaum's Outline of Geometry, 4th edition, 326 pages. McGraw-Hill Professional. ISBN 0071544127, 9780071544122. Online version accessed on 2009-08-08.
4. ^ Jonathan L. Gross, Jay Yellen (2006), Graph theory and its applications. 2nd edition, 779 pages; CRC Press. ISBN 158488505X, 9781584885054. Online version accessed on 2009-08-08.
5. ^ Definition of Radius at dictionary.reference.com. Accessed on 2009-08-08.

Categories: Geometry Geometry is the branch of mathematics dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible to proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, | Length Categories: Physical quantities | Dimension | Spacetime

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