[Math] How to find the length of diagonal of a rhombus

geometry

suppose,only the length of the side of a rhombus is given then how can I find the length of the diagonals?(without measuring the angle)
Is there any equation where the side is related to the diagonal only?

Best Answer

Consider any line segment AC whose length is $ < 2x $ ( $x$ being the length of the side of rhombus in question. )

From A cut an arc of radius $x$, then from C. They intersect say at B.

Now, you have an isosceles triangle ABC where $ AB = BC = x $

Reflect point B in the line segment AC to get a point outside the triangle, name it D.

Now we have : $ AB = AD = x $ and $ CB = CD = x $, so what do you have ?

We have a rhombus ABCD of side length $x$, whose diagonal can be anything between $0$ and $2x$.

If it were just one rhombus that existed you would not have arbitrary length diagonals to start with but only two distinct values the short and long diagonal.

So the rhombus in question is not unique !!

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