Congruent polygons

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Posted · Report post

Given that triangles have three sides and three angles we could share this information to another person to sketch our triangle. Of course, it is a known fact that we only would need to share three parts (sides/angles) at a minimum with someone else in order for them to construct our shape (e.g. Side, Side, and Side; Side, angle, and side). It is also known that to have someone build a quadrilateral for us, it would require five pieces of information. But what about a pentagon, hexagon,..., octagon, ... n-agon? Is there a generalizable formula as to the minimum amount of information needed for someone to construct your polygon unseen?

Shape: Minimum Information Needed:

Triangle 3

quadrilateral 5

Pentagon

Hexagon

.....

Octagon

n-agon

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Posted · Report post

You need at least one side length among your three pieces of information for a triangle. Three angles won't do.

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Posted · Report post

Four sides and their three included angles suffice.

You can construct an n-sided polygon given n-1 sides and their included angles.

The remaining side is specified by the remaining end points,

and the two remaining angles are what they are when those points are connected.

Thus 2n-3 is sufficient.

But is it necessary - i.e. minimal? Probably.

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Posted · Report post

A n-agon can be broken into n-2 triangles. Each triangle need three peces of information. But, after giving the information for the first triangle, every other triangle already has one peice of information, one side. Besides the first triangle, each triangle needs 2 pieces of information. The information needed for this would be 3+(2(n-3)) or 2n-3.

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Posted · Report post

1). Specifying numeric values for angles and sides is not enough. You can build up to 6 different convex quadrilaterals given four sides and one angle. Up to 3 different triangles given two sides and one angle.


To construct a polygon from given sides and angles, must know where they are in relation to one another.
2). Given n-1 angles for n-gon, we can figure out the n-th angle on our own. It does not represent any additional piece of information.
Or to put it differently, cannot leave more than 2 sides unspecified.
3). It would not hurt to ask whether the shape is not concaved. If it is -- must find out all the angles or the pairs of sides that are concaved before embarking on the figure building project.

Given the above, the formula is 2n - 3, as has been already noted.
That would be minimum and sufficient information to build an n-gon. I.e.:
Shape.......Minimum and sufficient
triangle ...........3
quadrilateral ...5
pentagon.........7
etc.

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