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# Encyclopedia

Circle illustration

In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter.[1]

More generally — in geometry, science, engineering, and many other contexts — the radius of something (e.g., a cylinder, a polygon, a mechanical part, a hole, or a galaxy) usually refers to the distance from its center or axis of symmetry 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 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.

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.

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.[4]

The name comes from Latin radius, meaning "ray" but also the spoke of a chariot wheel. The plural in English is radii (as in Latin), but radiuses can be used, though it rarely is.[5]

## Formulas for circles

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

$r = \frac{C}{2\pi}.$

The radius of a circle with area A is

$r= \sqrt{\frac{A}{\pi}}.$

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

$r=\frac{|P_1-P_3|}{2\sin\theta}$

where θ is the angle $\angle P_1 P_2 P_3.$

## Formulas for regular polygons

These formulas assume a regular polygon with n sides.

The radius can be computed from the side s by:

$r = R_n\, s$    where   $R_n = \frac{1}{2 \sin \frac{\pi}{n}} \quad\quad \begin{array}{r|ccr|c} n & R_n & & n & R_n\ \hline 2 & 0.50000000 & & 10 & 1.6180340- \ 3 & 0.5773503- & & 11 & 1.7747328- \ 4 & 0.7071068- & & 12 & 1.9318517- \ 5 & 0.8506508+ & & 13 & 2.0892907+ \ 6 & 1.00000000 & & 14 & 2.2469796+ \ 7 & 1.1523824+ & & 15 & 2.4048672- \ 8 & 1.3065630- & & 16 & 2.5629154+ \ 9 & 1.4619022+ & & 17 & 2.7210956- \end{array}$

## Formulas for hypercubes

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

$r = \frac{s}{2}\sqrt{d}.$

## 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.

# 1911 encyclopedia

Up to date as of January 14, 2010

### From LoveToKnow 1911

RADIUS, properly a straight rod, bar or staff, the original meaning of the Latin word, to which also many of the various meanings seen in English were attached; it was thus applied to the spokes of a wheel, to the semi-diameter of a circle or sphere and to a ray or beam of light, "ray" itself coming through the Fr. raie from radius. From this last sense comes "radiant," "radiation," and allied words. In mathematics, a radius is a straight line drawn from the centre to the circumference of a circle or to the surface of a sphere; in anatomy the name is applied to the outer one of the two bones of the fore-arm in man or to the corresponding bone in the fore-leg of animals. It is also used in various other anatomical senses in botany, ichthyology, entomology, &c. A further application of the term is to an area the extent of which is marked by the length of the radius from the point which is taken as the centre; thus, in London, for the purpose of reckoning the fare of hackneycarriages, the radius is taken as extending four miles in any direction from Charing Cross.

# Wiktionary

Up to date as of January 15, 2010

## German

### Noun

3. (geometry) radius (length of that line segment)

# Simple English

[[File:|thumb|The radius of a circle]] In geometry, the radius of a circle or sphere is the shortest connection between the center and the boundary. It is half of the diameter.

r= d ÷ 2

d= 2 x r = d= r + r

The relationship between the radius $\left\{\displaystyle r\right\}$ and the circumference $\left\{\displaystyle c\right\}$ of a circle is $~c = 2\pi r.$

The area $\left\{\displaystyle A\right\}$ of a circle of radius $\left\{\displaystyle r\right\}$ is $\left\{\displaystyle ~A = \pi r^2.\right\}$