Radius

For other uses, see Radius (disambiguation).

Circle illustration

In classical geometry, the radius of a circle or sphere is any line segment from its center to its perimeter the radius of a circle or sphere is the length of any such segment. By extension, the diameter is defined as twice the radius:

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

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. 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.

The name comes from Latin radius, meaning "ray" but also the spoke of a chariot wheel. The plural of radius can be either radii or the conventional English plural radiuses.

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

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

Radius from area

The radius of a circle with area A is

$r = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{2A}{\tau}}.$

The radius is half the diameter.

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 θ is the angle $\angle P_1 P_2 P_3.$

This formula uses the Sine Rule.

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   $R_n = \frac{1}{2 \sin \frac{\pi}{n}} \quad\quad \begin{array}{r|ccr|c} n & R_n & & n & R_n\\ \hline 2 & 0.50000000 & & 10 & 1.6180340- \\ 3 & 0.5773503- & & 11 & 1.7747328- \\ 4 & 0.7071068- & & 12 & 1.9318517- \\ 5 & 0.8506508+ & & 13 & 2.0892907+ \\ 6 & 1.00000000 & & 14 & 2.2469796+ \\ 7 & 1.1523824+ & & 15 & 2.4048672- \\ 8 & 1.3065630- & & 16 & 2.5629154+ \\ 9 & 1.4619022+ & & 17 & 2.7210956- \end{array}$

Formulas for hypercubes

Radius from side

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

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

See also

Atomic radius Bend radius Bohr radius Filling radius in Riemannian geometry Minimum railway curve radius Radius (bone)

Radius of gyration Radius of convexity Radius of convergence Radius of curvature Schwarzschild radius

External links

• Radius (PlanetMath.org website)