In classical geometry, a radius of a circle or sphere its 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, or a galaxy) usually refers to the distance from its center or axis of symmetry to its outermost points. 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 radius of a regular polygon (or polyhedron) is the distance from its center to any of its vertices; which is also its circumradius.

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 is also occasionally used.

Radius from circumference

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

$r\; =\; \backslash frac\{C\}\{2\backslash pi\}.$

Radius from area

The radius of a circle with area A is

$r=\; \backslash sqrt\{\backslash frac\{A\}\{\backslash pi\}\}.$

Radius from three points

To compute the radius of a circle going through three points P₁, P₂, P₃, the following formula can be used:

$r=\backslash frac\{2\backslash sin\backslash theta\}$

where θ is the angle $\backslash angle\; P\_1\; P\_2\; P\_3.$

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:

$r\; =\; R\_n\backslash ,\; s$    where   $R\_n\; =\; \backslash frac\{1\}\{2\; \backslash sin\; \backslash frac\{\backslash pi\}\{n\}\}\; \backslash quad\backslash quad$

Radius from side

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

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

See also

• Radius (bone)
• Radius of curvature
• Bend radius
• Radius of convexity
• Radius of convergence
• Radius of gyration
• Filling radius (Riemannian geometry)