Quadrilateral: Area Of A Convex QuadrilateralRelated Resources : Quadrilateral
Area Of A Convex QuadrilateralTrigonometric FormulasThe area can be expressed in trigonometric terms as pq \cdot \sin \theta, pq since θ is 90°. The area can be also expressed in terms of bimedians as where the lengths of the bimedians are m and n and the angle between them is φ. 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 Alternatively, we can write the area in terms of the sides and the intersection angle θ of the diagonals, so long as this angle is not 90°: \cdot \left a^2 + c^2  b^2  d^2 \right .\tan \theta\cdot \left a^2  b^2 \right. Another area formula including the sides a, b, c, d is where x is the distance between the midpoints of the diagonals and φ is the angle between the bimedians. Nontrigonometric FormulasThe following two formulas express the area in terms of the sides a, b, c, d, the semiperimeter s, and the diagonals p, q: The area can also be expressed in terms of the bimedians m, n and the diagonals p, q: In fact, any three of the four values m, n, p, and q suffice for determination of the area, since in any quadrilateral the four values are related by $p^2+q^2=2m^2+n^2.$ The list applies to the most general cases, and excludes named subsets.
Note 1: The most general trapezoids and isosceles trapezoids do not have perpendicular diagonals, but there are infinite numbers of nonsimilar trapezoids and isosceles trapezoids that do have perpendicular diagonals and are not any other named quadrilateral. Note 2: In a kite, one diagonal bisects the other. The most general kite has unequal diagonals, but there is an infinite number of nonsimilar kites in which the diagonals are equal in length and the kites are not any other named quadrilateral. Length Of The DiagonalsOther, more symmetric formulas for the length of the diagonals, are Generalizations Of The Parallelogram Law And Ptolemy S TheoremIn any convex quadrilateral ABCD, the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals plus four times the square of the line segment connecting the midpoints of the diagonals. Thus where x is the distance between the midpoints of the diagonals. Other Metric Relationsa^2 & 0 & b^2 & q^2 & 1 \ p^2 & b^2 & 0 & c^2 & 1 \ d^2 & q^2 & c^2 & 0 & 1 \ Related Categories66 Related Topics about Quadrilateral
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arithmetic mean ..
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centroid ..
circumcenter ..
circumscribed circle ..
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cuttheknot ..
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degrees of arc ..
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if and only if ..
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interior angles ..
isosceles trapezium ..
law of cosines ..
line segment ..
midpoint ..
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parallel ..
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right angle ..
simple ..
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tangent ..
taxonomy ..
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tile the plane ..
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