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Bretschneider's Formula

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Given a general quadrilateral with sides of lengths a, b, c, and d, the area is given by

K=1/4sqrt(4p^2q^2-(b^2+d^2-a^2-c^2)^2)
(1)
=sqrt((s-a)(s-b)(s-c)(s-d)-1/4(ac+bd+pq)(ac+bd-pq))
(2)

(Coolidge 1939; Ivanov 1960; Beyer 1987, p. 123) where p and q are the diagonal lengths and s is the semiperimeter. While this formula is termed Bretschneider's formula in Ivanoff (1960) and Beyer (1987, p. 123), this appears to be a misnomer. Coolidge (1939) gives the second form of this formula, stating "here is one [formula] which, so far as I can find out, is new," while at the same time crediting Bretschneider (1842) and Strehlke (1842) with "rather clumsy" proofs of the related formula

K= sqrt((s-a)(s-b)(s-c)(s-d)-abcdcos^2[1/2(A+B)])
(3)

(Bretschneider 1842; Strehlke 1842; Coolidge 1939; Beyer 1987, p. 123), where A and B are two opposite angles of the quadrilateral.

QuadrilateralVectors

"Bretschneider's formula" can be derived by representing the sides of the quadrilateral by the vectors a, b, c, and d arranged such that a+b+c+d=0 and the diagonals by the vectors p and q arranged so that p=b+c and q=a+b. The area of a quadrilateral in terms of its diagonals is given by the two-dimensional cross product

K=1/2|pxq|,
(4)

which can be written

K^2=1/4(pxq)·(pxq),
(5)

where u·v denotes a dot product. Making using of a vector quadruple product identity gives

K=1/2sqrt((p·p)(q·q)-(p·q)^2)
(6)
=1/2sqrt(p^2q^2-(p·q)^2).
(7)

But

2(p·q)=2(b+c)·(a+b)
(8)
=-2b·(c+d)+2c·(a+b)
(9)
=2a·c-2b·d
(10)
=(a+c)·(a+c)-a·a-c·c-(b+d)·(b+d)+b·b+d·d
(11)
=b^2-a^2+d^2-c^2.
(12)

Plugging this back in then gives the original formula (Ivanoff 1960).

See also

Brahmagupta's Formula, Heron's Formula, Quadrilateral

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References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987.Bretschneider, C. A. "Untersuchung der trigonometrischen Relationen des geradlinigen Viereckes." Archiv der Math. 2, 225-261, 1842.Coolidge, J. L. "A Historically Interesting Formula for the Area of a Quadrilateral." Amer. Math. Monthly 46, 345-347, 1939.Dostor, G. "Propriétés nouvelle du quadrilatère en général avec application aux quadrilatéres inscriptibles, circonscriptibles." Arch. Math. Phys. 48, 245-348, 1868.Hobson, E. W. A Treatise on Plane and Advanced Trigonometry. New York: Dover, pp. 204-205, 1957.Ivanoff, V. F. "Solution to Problem E1376: Bretschneider's Formula." Amer. Math. Monthly 67, 291-292, 1960.Strehlke, F. "Zwei neue Sätze vom ebenen und shparischen Viereck und Umkehrung des Ptolemaischen Lehrsatzes." Archiv der Math. 2, 33-326, 1842.

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Bretschneider's Formula

Cite this as:

Weisstein, Eric W. "Bretschneider's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BretschneidersFormula.html

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