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.

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). 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. 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.

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.

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


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

Radius from area
The radius of a circle with area A is
 * $$r= \sqrt{\frac{A}{\pi}}.$$

Radius from three points
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 &theta; is the angle $$ \angle P_1 P_2 P_3.$$

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:


 * $$r = R_n\, s$$   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} $$

Radius from side
The radius of a d-dimensional hypercube with side s is
 * $$ r = \frac{s}{2}\sqrt{d}.$$