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I have an irregular quadrilateral. I know the length of three sides (a, b and c) and the length of the two diagonals (e and f). All angles are unknown How do I calculate the length of the 4th side (d)?

Thank you for your help. Regards,

Mo

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Toby Mak
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user453690
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    Explain the downvotes, you should try and comment with the user to see if they can add context and more information. This is quite an interesting question, it makes no sense to downvote it if they can try. – Toby Mak Jun 09 '17 at 08:30

2 Answers2

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Hint : You could try re-arranging the Cosine Rule: $a^2 = c^2+b^2-2bc\cos A$ to try and find some of the angles of the triangles.

Plus Twenty
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You could use Brahmagupta's formula which is $\sqrt{s(s-a)(s-b)(s-c)(s-d)}$ where $s$ is the semiperimeter $(\frac{1}{2})(a+b+c+d)$ only if the quadrilateral is cyclic, in which $e$ and $f$ are equal.

The proof uses angles and the law of cosines, so the side lengths a quadrilateral with no constraints on its angles cannot be determined. Take this quadrilateral and this quadrilateral which have all four side lengths measuring $\sqrt2$ units, which is another piece of evidence supporting my proof.

Toby Mak
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