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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] 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.

For regular polygons, the radius is the same as its circumradius.[2] 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.[3]

The name comes from Latin radius, meaning "ray" but also the spoke of a chariot wheel. [4] The plural of radius is radii.

## Formulas for circles

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

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

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 $\angle P_1 P_2 P_3.$

This formula uses the Sine Rule.

## 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. ^ 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.
3. ^ 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.
4. ^ Definition of Radius at dictionary.reference.com. Accessed on 2009-08-08.

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