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biggest little polygon
In geometry, the biggest little polygon for a number ''n'' is the ''n''-sided polygon that has diameter one (that is, every two of its points are within unit distance of each other) and that has the largest area among all diameter-one ''n''-gons. One non-unique solution when ''n'' = 4 is a square, and the solution is a regular polygon when ''n'' is an odd number, but the solution is irregular otherwise.
==Quadrilaterals==
For ''n'' = 4, the area of an arbitrary quadrilateral is given by the formula ''S'' = ''pq'' sin(''θ'')/2 where ''p'' and ''q'' are the two diagonals of the quadrilateral and ''θ'' is either of the angles they form with each other. In order for the diameter to be at most 1, both ''p'' and ''q'' must themselves be at most 1. Therefore, the quadrilateral has largest area when the three factors in the area formula are individually maximized, with ''p'' = ''q'' = 1 and sin(''θ'') = 1. The condition that ''p'' = ''q'' means that the quadrilateral is an equidiagonal quadrilateral (its diagonals have equal length), and the condition that sin(''θ'') = 1 means that it is an orthodiagonal quadrilateral (its diagonals cross at right angles). The quadrilaterals of this type include the square with unit-length diagonals, which has area 1/2. However, infinitely many other orthodiagonal and equidiagonal quadrilaterals also have diameter 1 and have the same area as the square, so in this case the solution is not unique.〔. As cited by .〕
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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