Area of a Parallelogram
The area of a parallelogram equals its base times its perpendicular height, derived by rearranging the shape into a rectangle.
Formula
A = b \times h
Definition
The area of a parallelogram with base $b$ and perpendicular height $h$ is $A = bh$, where $h$ is the straight-up distance between the two parallel bases, not the length of the slant side. This is identical to the rectangle formula because a parallelogram can be sheared into a rectangle without changing its area: cut a triangle off one end and slide it to the other, and you get a rectangle with the same base and height. In vector form, the area of the parallelogram spanned by $u = (u_1, u_2)$ and $v = (v_1, v_2)$ is $|\det([u, v])| = |u_1 v_2 - u_2 v_1|$, and in 3-D the area spanned by vectors $u$ and $v$ is $|u \times v|$, the geometric interpretation of the cross product.
Example
A parallelogram with base $8$ cm and height $5$ cm has $A = 8 \times 5 = 40$ cm². A parallelogram with base $12$ m, slant side $7$ m, and height $5$ m has area $12 \times 5 = 60$ m$^2$; the $7$ m slant side is never used in the calculation. For vectors $u = (3, 1)$ and $v = (1, 4)$, the parallelogram they span has area $|3 \cdot 4 - 1 \cdot 1| = |12 - 1| = 11$ square units.
Key Insight
A parallelogram is a "slanted" rectangle: no matter how much it leans, the area depends only on the base and the straight-up height, never on the slant. Shear transformations preserve area, so a parallelogram and a rectangle with the same base and height are always equal in area. The determinant as a signed area of a parallelogram underlies the change-of-variables formula in multivariable integration, where the Jacobian determinant measures how areas scale under a substitution.