In classical geometry, a radius (pl.: radii or radiuses) 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 radius of a regular polygon is the line segment or distance from its center to any of its vertices. The name comes from the Latin radius, meaning ray but also the spoke of a chariot wheel. The typical abbreviation and mathematical symbol for radius is R or r. By extension, the diameter D is defined as twice the radius:

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. The inradius of a regular polygon is also called the 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.

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

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

The radius of a circle with area A is

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

where θ is the angle ∠P₁P₂P₃. This formula uses the law of sines. If the three points are given by their coordinates (x₁,y₁), (x₂,y₂), and (x₃,y₃), the radius can be expressed as