Circle illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta.

In classical geometry, the radius of a circle or sphere is the length of a line segment from its center to its perimeter. 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]

$d \doteq 2r \quad \Rightarrow \quad r = \frac{d}{2}.$

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.

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.[5]

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 $\angle P_1 P_2 P_3.$

This formula uses the Sine Rule.

If the three points are given by their coordinates $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$, one can also use the following formula :

$r={\frac {\sqrt{ \left( \left( {\it x_2}-{\it x_1} \right) ^{2}+ \left( {\it y_2}-{\it y_1} \right) ^{2} \right) \left( \left( {\it x_2}-{\it x_3} \right) ^{2}+ \left( {\it y_2}-{\it y_3} \right) ^{2} \right) \left( \left( {\it x_3}-{\it x_1} \right) ^{2}+ \left( {\it y_3}-{\it y_1} \right) ^{2} \right)} }{ 2 \left| {\it x_1}\,{\it y_2}+{\it x_2}\,{\it y_3}+{\it x_3}\,{\it y_1}-{\it x_1}\,{\it y_3}-{\it x_2}\,{\it y_1}-{\it x_3}\,{\it y_2} \right| }}$

## 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

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