Bretschneider's Quadrilateral Area Formula
The following remarkable general formula for the area of any quadrilateral ABCD (convex, concave, crossed) is attributed to Bretschneider (1842):
where a = AB, b = BC, c = CD, d = DA and s is the semiperimeter (a + b + c + d)/2.
Explore
In the dynamic sketch below a crossed cyclic quadrilateral ABCD is shown (with directed vertex angles as indicated).
1) Click on the 'Show Area' button, which shows the area as determined by WebSketchpad as well as the calculated value according to Bretschneider's formula above.
2) Keeping ABCD crossed, check further by dragging any of the vertices to see if these measurements and calculations always agree.
3) Click on the 'Show Angle Sums' button.
4) If ABCD remains crossed, what do you notice about the sum of the opposite (or alternate) angles at A and C (and at B and D)?
5) What does Bretschneider's formula reduce to if ∠A + ∠C = 360°?
6) While ABCD is still a crossed cyclic quadrilateral, click on the 'Show Brahmagupta' button, which shows Brahmagupta's area formula.
7) What do you notice in 6) above?
8) Now drag vertex B clockwise along the circumference of the circle until it moves past vertex C so that ABCD becomes a convex, cyclic quadrilateral.
9) What do you now notice regarding Brahmagupta's area formula?
10) Click on the 'Link to general quad area' button to navigate to a new sketch where the vertex A has been detached from the circle so that ABCD is now a general quadrilateral.
11) Explore Bretschneider's formula for a general quadrilateral further by dragging ABCD into convex and concave shapes.
Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula
Brahmagupta's Formula
In 6)-9) above, you should've observed that Brahmagupta's formula only holds for a convex cyclic quadrilateral ABCD. Note that since the opposite angles of a convex cyclic quadrilateral are supplementary, Bretschneider's formula reduces to Brahmagupta's formula, which is:
where a = AB, b = BC, c = CD, d = DA and s is the semiperimeter (a + b + c + d)/2.
Historical Note: Brahmagupta's famous formula is named after an Indian mathematician and astronomer (c. 598 - 670) by the same name.
Heron's Formula
If in Brahmagupta's formula we let two of the vertices of a convex cyclic quadrilateral ABCD coincide, say vertex D with C, to obtain a triangle ABC, the formula reduces to Heron's formula for the area of a triangle, namely:
where a = AB, b = BC, c = CA, and s is the semiperimeter (a + b + c)/2.
Historical Note: Heron (or Hero) of Alexandria, Egypt ( AD 10 - AD 75) was not only a famous mathematician, but also a mechanical engineer of his times.
Challenge
The above formulae are often useful in the proof of other advanced geometry results, and often also feature in mathematical olympiad competitions.
12) Can you prove or derive the above formulae by yourself?
Related Links
Area Formula for Quadrilateral in terms of its Diagonals
Finding the Area of a Crossed Quadrilateral
Crossed Quadrilateral Properties
The Equi-partitioning Point of a Quadrilateral
International Mathematical Talent Search (IMTS) Problem Generalized
A Geometric Paradox Explained (Another variation of an IMTS problem)
Another parallelogram area ratio
Area Parallelogram Partition Theorem: Another Example of the Discovery Function of Proof
Area ratios of some polygons inscribed in quadrilaterals and triangles
A generalization of the Cyclic Quadrilateral Angle Sum theorem
Crossed Cyclic Hexagon Alternate Angles Sum Theorem
The theorem of Hippocrates (470 – c. 410 BC)
Feynman's Triangle: Some Generalizations & Variations
Maximum area of quadrilateral problem
External Links
Biography of Heron of Alexandria (MacTutor)
Biography of Brahmagupta (MacTutor)
Bretschneider's Formula (WolfRam Math World)
Brahmagupta's Formula (WolfRam Math World)
Heron's Formula (WolfRam Math World)
Heron's Formula (Wikipedia)
Directed Angle (WolfRam Math World)
How to Use Directed Angles by Evan Chen (2015, pdf)
SA Mathematics Olympiad
Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Created with WebSketchpad by Michael de Villiers, 20 October 2024.