Area of a Parallelogram

A parallelogram is a quadrilateral whose opposite sides are parallel. Its area is simply the base multiplied by the height measured perpendicular to that base.

• $S$ — the area
• $b$ — the length of the base
• $h$ — the height, measured perpendicular to the base

Example

With the defaults, a base of $b = 6$ and a height of $h = 4$:

The area is 24.

Where the formula comes from

Slice a right triangle off one end of the parallelogram, cutting along the height, and slide it across to the other end. The pieces reassemble into a rectangle of height $h$ and width $b$. Nothing was added or thrown away, so the area is unchanged, and it equals $b h$, exactly as for the rectangle.

Watch out

The classic mistake is to use the slanted side where the height belongs. The height is the perpendicular distance between the two parallel sides, so it is always shorter than the slanted side; the two coincide only when the shape is a rectangle. If you know the slanted side $a$ and the angle $\theta$ it makes with the base, first compute $h = a \sin\theta$. Either pair of parallel sides may serve as the base, as long as you use the height that goes with it.