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Quadrilateral: Area Of A Convex Quadrilateral


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Area Of A Convex Quadrilateral

Trigonometric Formulas

The 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.

Non-trigonometric Formulas

The 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.


Quadrilateral Bisecting diagonals Perpendicular diagonals Equal diagonals
Trapezoid No See note 1 No
Isosceles trapezoid No See note 1 Yes
Parallelogram Yes No No
Kite See note 2 Yes See note 2
Rectangle Yes No Yes
Rhombus Yes Yes No
Square

Note 1: The most general trapezoids and isosceles trapezoids do not have perpendicular diagonals, but there are infinite numbers of non-similar 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 non-similar kites in which the diagonals are equal in length and the kites are not any other named quadrilateral.

Length Of The Diagonals

Other, more symmetric formulas for the length of the diagonals, are

Generalizations Of The Parallelogram Law And Ptolemy S Theorem

In 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 Relations

a^2 & 0 & b^2 & q^2 & 1 \

p^2 & b^2 & 0 & c^2 & 1 \

d^2 & q^2 & c^2 & 0 & 1 \


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