Parallelogram
A parallelogram is a quadrilateral with two pairs of parallel sides, with opposite sides equal and opposite angles equal.
Formula
\text{Area} = \text{base} \times \text{height}
Definition
A parallelogram is a four-sided shape where both pairs of opposite sides are parallel and equal in length, with opposite angles equal too; consecutive angles are supplementary and diagonals bisect each other. Area $=$ base $\times$ height, using the perpendicular height between the parallel sides, not the slanted side length. With adjacent sides as vectors $u$ and $v$, the area is $|u \times v|$, the diagonals are $u+v$ and $u-v$, and the parallelogram law states $|d_1|^2+|d_2|^2 = 2(|u|^2+|v|^2)$.
Example
A leaning stack of books or the diamond on a baseball field is shaped like a parallelogram; a rectangle is a special parallelogram with right-angle corners. With base $10$, slant side $8$, and height $6$: area $=10\times6=60$, not $10\times8$. For $u=(3,0)$ and $v=(1,4)$: area $=|3\cdot4-0\cdot1|=12$, diagonals $u+v=(4,4)$ with length $4\sqrt{2}$ and $u-v=(2,-4)$ with length $2\sqrt{5}$, and $32+20=52=2(9+17)$, confirming the parallelogram law.
Key Insight
The area formula works because a parallelogram can be rearranged into a rectangle of the same base and height by moving a right triangle from one end to the other, a visual proof that area is preserved despite the shape change. The parallelogram law characterizes inner product spaces: a normed space has a dot product if and only if the parallelogram law holds for it, an algebraic identity with roots in this simple diagonal-side relationship, and the symmetry group of a general parallelogram is just $\mathbb{Z}_2$ (a $180^\circ$ rotation).