Find area of irregular 6 sided polygon (only by sdies)

Question

I have to find area of 6 sided irregular polygon, and I have no angle here. Is there a trick to find the area with this image information?

I got stuck on this problem. There is some online calculator for this reason but they need to out information as triangles.( But I do not know $AC,CE ,...$

Thanks for any help in advance.

It's impossible without knowing the angles or the diagonals. You could find the angles with a goniometer on your picture. — Intelligenti pauca, Jun 30 '22 at 17:21
Just imagine to build a model of that figure with six hinged bars. It's obvious your model can be deformed and its area changed. — Intelligenti pauca, Jun 30 '22 at 17:29
There is no unique solution. Think of a simpler example, a quadrilateral with all sides equal to 1. Then you can get a square, if the angles are $90^\circ$, but you can get a rhombus. The area of the rhombus is dependent on the smallest (or largest) angle. — Andrei, Jun 30 '22 at 17:56
@user2661923 I'd suggest you to draw that figure with GeoGebra: that'll be better than any theoretical explanation. — Intelligenti pauca, Jun 30 '22 at 18:31
@Intelligentipauca Good idea. After re-considering, I deleted all of my previous comments. — user2661923, Jun 30 '22 at 19:15
Does this answer your question? Does specifying the lengths of the sides of a polygon completely fix its shape (area and angles)? — dxiv, Jun 30 '22 at 20:55

Narasimham · Answer 1 · 2022-06-30T20:38:46.303

1

For full triangulation we need $2n-3$ dimensions in a plane. ( $n$ is number of vertices or joints or nodes). Less than this number we have a mechanism with the vertices as hinges.

When $n=6$ we need 9, so three more dimensions (side lengths and/or angles) are needed without which a construction is impossible.

Static indeterminacy in Trusses

$$ m=2 j-3 $$

This relation determines static indeterminacy in pin jointed trusses. Less than this $j$ the truss is statically indeterminate as in the given case of a mechanism ($ 2j-3=9$). More than this it is redundant, and is said to define the degree of freedom.

Unless the degree of freedom is zero area cannot be found.

edited Jun 30 '22 at 20:38

answered Jun 30 '22 at 18:02

Narasimham

40,495

Very interesting answer. Can you recommend a source that discusses that full triangulation requires $(2n-3)$ dimensions in the plane? – user2661923 Jun 30 '22 at 19:18
It is derived easily in triangulated trusses. Each extra joint $j$ or vertex gives two extra members. So it is of the form $2j +k$ where $k$ is found from special case of a triangle. – Narasimham Jun 30 '22 at 20:35

Find area of irregular 6 sided polygon (only by sdies)

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