areageometryquadrilateral
Below the parallelogram is obtained from square by stretching the top side while fixing the bottom.
Since area of parallelogram is base times height, both square and parallelogram have the same area.
This is true no matter how far I stretch the top side.
In below figure it is easy to see why both areas are same.
But it's not that obvious in first two figures. Any help seeing why the area doesn't change in first figure?
The question is where do you start with the definition of "area". As far as I know, there are two options:
Develop a general notion of a measure in the plane, say, in the spirit of the Lebesgue integral. Then the formula for the measure of a rectangle comes as a part of the definition of the measure. The hard part of the theory is to construct a well-defined set-function on a certain algebra of subset of the plane (or $n$-space, there is no fundamental difference between the two constructions).
You can also start with axioms of Euclidean geometry (which do not include anything about areas!) and then develop a concept of area for general (not only convex) polygonal regions in the plane. If your objective is to present something that high school students or typical undergraduate students (in a Euclidean geometry class) can comprehend, then this is a way to go. Take a look at the book
E. Moise, "Elementary geometry from advanced standpoint", especially section 13.5, where he shows that area of any square (with side $a$) is $a^2$, starting merely with the normalization that area of the unit square is 1. Along the way, Moise develops the notion of area of planar polygons from scratch. The most difficult part of his proof (that area exists) is to show that area of a polygon does not depend on triangulation.
To conclude: all you need is the set of (Hilbert's) axioms of Euclidean geometry (since, as Hilbert observed, Euclid's set of axioms is incomplete). You also need a notion of real numbers (which Euclid did not have either), namely, ordered field axioms and the completeness axiom.
You can see what's going on without algebra. Given sides $a$ and $b$ there will always be two triangles with the same area for each angle $\theta$ at the intersection of $a$ and $b$ (except at the maximum area).
Another way to see this analytically is to note that the area of the triangle is $ab \sin(\theta)/2$.
Best Answer
Behold, $\phantom{proof without words}$