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Since the diameter is twice the radius, the "missing" part of the diameter is (2r − x) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (2r − x)x = (y / 2) 2. Solving for r, we find the required result.
The diameter is always attained by two points of the convex hull of the input. A trivial brute-force search can be used to find the diameter of n {\displaystyle n} points in time O ( n 2 ) {\displaystyle O(n^{2})} (assuming constant-time distance evaluations) but faster algorithms are possible for points in low dimensions.
Jung's theorem provides more general inequalities relating the diameter to the radius. [5] The isodiametric inequality or Bieberbach inequality , a relative of the isoperimetric inequality , states that, for a given diameter, the planar shape with the largest area is a disk, and the three-dimensional shape with the largest volume is a sphere.
A diameter of an ellipse is any line passing through the centre of the ellipse. [2] Half of any such diameter may be called a semidiameter, although this term is most often a synonym for the radius of a circle or sphere. [3] The longest diameter is called the major axis.
The hydraulic diameter is the equivalent circular configuration with the same circumference as the wetted perimeter. The area of a circle of radius R is . Given the area of a non-circular object A, one can calculate its area-equivalent radius by setting = or, alternatively:
Consider a circle in with center at the origin and radius . Gauss's circle problem asks how many points there are inside this circle of the form ( m , n ) {\displaystyle (m,n)} where m {\displaystyle m} and n {\displaystyle n} are both integers.
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The radius of a circle is perpendicular to the tangent line through its endpoint on the circle's circumference. Conversely, the perpendicular to a radius through the same endpoint is a tangent line. The resulting geometrical figure of circle and tangent line has a reflection symmetry about the axis of the radius.