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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°.

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 expresses 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:

### Area Inequalities

If a convex quadrilateral has the consecutive sides a, b, c, d and the diagonals p, q, then its area K satisfies

p^2+q^2 with equality only if the diagonals are perpendicular and equal.

with equality only for a rectangle.

### Related Categories

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