Board 8 > Math peeps, can you solve the area of a trapezoid if you only know sides?

(+) Yeah! (+)

No angles, sides are 35, 35, 45, and 12
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yes,

you can also calculate the angle if you want to. The main trick is to think of it as two triangles on both sides of a rectangle. Calculate the third side of one of the triangles will give you the height of the rectangle, aka the height of the trapezoid.

Then just use the formula:

height * ((top+bottom)/2)
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Wait it's not a trapezoid. The sides are not parallel.

Is there still a way?
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Need some angle information then
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But wouldn't there only be one set of angles that would work?
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No. That would be true for a triangle, but not for a quadrilateral.

As a simple counterexample, you can considering bending a square into a rhombus (which will have a smaller area, as you can clearly see in the limit of totally flattening it out).
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Okay, yeah that makes sense.
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Are the two 35's adjacent? I'm assuming if you thought this was a trapezoid that they are not.

Menji posted...

But wouldn't there only be one set of angles that would work?

I gave this a moment's thought to consider if LotM's counterexample applies in general, or only in limited cases. I eventually concluded that it's a general statement. Split the shape with a diagonal. If you know that diagonal, both "halves" of the quad are specific triangles by SSS. However, the length of the diagonal is not unique, and depends on how wide the angles open. So yeah, we need at least one angle. If you say there's a right angle somewhere, that'll do the trick.
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They are adjacent
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