$\begingroup$By definition parallelograms are special type of trapezoids. Given a trapezoid with known sides one can calculate its area according to wikipedia. But in order to calculate the area of a parallelogram you need to know its height. I find this paradoxical. Can someone help clarifying this?
Edit: What I think I want to know is why trapezoid in general are solid(if that's the right word) but parallelograms are not. Rhombi are parallelograms. I can imagine changing a rhombus's diagonals (hence its area) without changing its sides.
$\endgroup$2 Answers
$\begingroup$There is no paradox. The area of a trapezoid can be given by $$A = (b_1 + b_2)h/2,$$ where $b_1$ and $b_2$ are the lengths of the two parallel sides, and $h$ is the distance (height) between the parallel sides.
In a parallelogram, we have the special case $b_1 = b_2 = b$, thus the formula reduces to $$A = (2b)h/2 = bh.$$
$\endgroup$2$\begingroup$Note that the Wikipedia formula involves the denominator $|b-a|$ which becomes $0$ in the case of the parallelogram; i.e. the formula only applies for non-parallelogram trapezoids.
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