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

In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the 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 mathematical variable name for radius is r. By extension, the diameter d is defined as twice the radius: [3]

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

If an object does not have a 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

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

## Formula

For many geometrical figures, the radius has a well-defined relationship with other measures of the figure.

### Circles

The radius of a circle with area A is

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

The radius of the circle that passes through the three non- collinear points P1, P2 and P3 is given by

${\displaystyle r={\frac {|{\vec {OP_{1}}}-{\vec {OP_{3}}}|}{2\sin \theta }},}$

where θ is the angle ${\displaystyle \angle P_{1}P_{2}P_{3}.}$ This formula uses the law of sines. If the three points are given by their coordinates ${\displaystyle (x_{1},y_{1})}$, ${\displaystyle (x_{2},y_{2})}$ and ${\displaystyle (x_{3},y_{3})}$, the radius can be expressed as

${\displaystyle r={\frac {\sqrt {((x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2})((x_{2}-x_{3})^{2}+(y_{2}-y_{3})^{2})((x_{3}-x_{1})^{2}+(y_{3}-y_{1})^{2})}}{2|x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{1}y_{3}-x_{2}y_{1}-x_{3}y_{2}|}}.}$

### Regular polygons

The radius of a regular polygon with n sides of length s is given by ${\displaystyle r=R_{n}\,s}$, with ${\displaystyle R_{n}=1/\left(2\sin {\frac {\pi }{n}}\right):}$

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

### Hypercubes

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

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