Parallelogram Area
A = b·hP = 2·(b+s)The area of a parallelogram is base times height — exactly the same formula as a rectangle.
The reason is geometric: cut the triangle off one end and fit it onto the other, and the parallelogram becomes a rectangle of the same base and height. The slant does not change the area.
The perimeter, on the other hand, depends on the slanted side: it is twice the sum of the base and that side.
This is the central mistake in this calculation, and the same one that occurs with trapezoids. The height is the perpendicular distance between the two parallel bases.
The slanted side is always longer than the height, except when the parallelogram is a rectangle and the two coincide. Using the side in place of the height always overestimates the area.
The more the figure leans, the larger the error: a heavily slanted parallelogram can have a side several times longer than its actual height.
Because area does not depend on the slant, a 10 by 4 rectangle and a heavily slanted parallelogram with base 10 and height 4 have exactly the same area.
What differs is the perimeter: the slanted parallelogram has a longer outline, because its oblique sides are longer.
It is the same distinction that appears with rectangles: area and perimeter measure different things and do not move together.
Frequently asked questions
The perpendicular distance between the two parallel bases. It is not the slanted side — using that always yields an area larger than the real one.
Because cutting the triangle off one end and fitting it onto the other turns the parallelogram into a rectangle of the same base and height. The slant does not change the area.
Yes. A parallelogram only requires opposite sides to be parallel; a rectangle adds the requirement of right angles.
No, only for the perimeter. The area depends solely on base and height.