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The area K of an orthodiagonal quadrilateral equals one half the product of the lengths of the diagonals p and q: [7] 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. [ 5 ]
Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities; List of volume formulas – Quantity of three-dimensional space
More generally, if the quadrilateral is a rectangle with sides a and b and diagonal d then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then d 2, the right hand side of Ptolemy's relation is the sum a 2 + b 2.
Bretschneider's formula generalizes Brahmagupta's formula for the area of a cyclic quadrilateral, which in turn generalizes Heron's formula for the area of a triangle.. The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals e and f to give [2] [3]
The only other quadrilateral with such a property is that of a three by six rectangle. 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
Now the smaller piece can be detached, rotated, and placed to the right of the larger piece. In this new arrangement, shown in the last diagram below, the two pieces together form a rectangle, whose width is + and whose height is . This rectangle's area is (+) (). Since this rectangle came from rearranging the original figure, it must have the ...
This reduces to Brahmagupta's formula for the area of a cyclic quadrilateral—when A + C = 180°. Another area formula in terms of the sides and angles, with angle C being between sides b and c, and A being between sides a and d, is = + .
The diagonals of a cube with side length 1. AC' (shown in blue) is a space diagonal with length , while AC (shown in red) is a face diagonal and has length .. In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge.
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