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The formula for the perimeter of a rectangle The area of a rectangle is the product of the length and width. If a rectangle has length and width , then: [11] it has area =; it has perimeter = + = (+); each diagonal has length = +; and
The area K of an orthodiagonal quadrilateral equals one half the product of the lengths of the diagonals p and q: [8] K = p q 2 . {\displaystyle K={\frac {pq}{2}}.} Conversely, any convex quadrilateral where the area can be calculated with this formula must be orthodiagonal. [ 6 ]
That is, the area of the rectangle is the length multiplied by the width. As a special case, as l = w in the case of a square, the area of a square with side length s is given by the formula: [1] [2] A = s 2 (square). The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a ...
The dual theorem states that of all quadrilaterals with a given area, the square has the shortest perimeter. The quadrilateral with given side lengths that has the maximum area is the cyclic quadrilateral. [43] Of all convex quadrilaterals with given diagonals, the orthodiagonal quadrilateral has the largest area.
Given a triangle with sides of length a, b, and c, if a 2 + b 2 = c 2, then the angle between sides a and b is a right angle. For any three positive real numbers a , b , and c such that a 2 + b 2 = c 2 , there exists a triangle with sides a , b and c as a consequence of the converse of the triangle inequality .
In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the term square to mean raising to the second power. The area can also be calculated using the diagonal d according to =. In terms of the circumradius R, the area of a square is
If the quadrilateral is rectangle, then equation simplifies further since now the two diagonals are of equal length as well: 2 a 2 + 2 b 2 = 2 e 2 {\displaystyle 2a^{2}+2b^{2}=2e^{2}} Dividing by 2 yields the Euler–Pythagoras theorem:
The diagonals divide the polygon into 1, 4, 11, 24, ... pieces. [ a ] For a regular n -gon inscribed in a circle of radius 1 {\displaystyle 1} , the product of the distances from a given vertex to all other vertices (including adjacent vertices and vertices connected by a diagonal) equals n .