Parallelogram
GeometryA 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 (never meet) and equal in length. The opposite angles are also equal.
Example
A leaning stack of books, a slanted piece of bread, or the diamond on a baseball field are shaped like parallelograms. A rectangle is a special parallelogram where all corners are right angles.
Key Insight
To find the area of a parallelogram, use base $\times$ height - but the height is the perpendicular distance between the parallel sides, NOT the slanted side length. The height is always measured straight up, not along the slant.
Definition
A parallelogram is a quadrilateral with both pairs of opposite sides parallel and congruent. Properties: opposite angles are congruent, consecutive angles are supplementary, diagonals bisect each other. Area $=$ base $\times$ height (perpendicular height, not slant height).
Example
Parallelogram with base $10$, slant side $8$, and height $6$: Area $= 10 \times 6 = 60$ (not $10 \times 8$). The diagonals bisect each other but are not necessarily equal or perpendicular (unlike a rectangle or rhombus).
Key Insight
The area formula base $\times$ height works because a parallelogram can be rearranged into a rectangle of the same base and height: cut a right triangle from one end and move it to the other. This visual proof shows area is preserved despite the shape change.
Definition
A parallelogram with adjacent sides as vectors $u$ and $v$ has area $|u \times v|$ (magnitude of the cross product). The diagonals are $u + v$ and $u - v$; the sum of squares of diagonals equals twice the sum of squares of sides: $|d_1|^2 + |d_2|^2 = 2(|u|^2 + |v|^2)$ (parallelogram law). The symmetry group of a general parallelogram is $\mathbb{Z}_2$ ($180^\circ$ rotation only).
Example
Parallelogram with $u = (3, 0)$ 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 $2\sqrt{5}$. Check: $32 + 20 = 52 = 2(9+17) = 52$. Confirmed.
Key Insight
The parallelogram law ($|u+v|^2 + |u-v|^2 = 2|u|^2 + 2|v|^2$) characterizes inner product spaces: a normed space is an inner product space (has a dot product) iff the parallelogram law holds. This algebraic identity has geometric roots in the diagonal-side relationship of the parallelogram.