# Radius

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]}

## Contents

## Formulas for circles

### Radius from circumference

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

### Radius from area

The radius of a circle with area *A* is

### Radius from three points

To compute the radius of a circle going through three points *P*_{1}, *P*_{2}, *P*_{3}, 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

## References

- ↑ Definition of radius at mathwords.com. Accessed on 2009-08-08.
- ↑
Robert Clarke James, Glenn James (1992),
*Mathematics dictionary*. 548 pages, Springer ISBN 0412990415, 9780412990410 - ↑
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. - ↑
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. - ↑ Definition of Radius at dictionary.reference.com. Accessed on 2009-08-08.