Area of a Parallelogram

Cut a right-angled triangle off one end of a parallelogram and slide it to the other end. A rectangle appears.

Nothing was gained or lost

The piece moved, so the area is unchanged. The parallelogram and the rectangle cover the same amount.

So the formula is the rectangle

Area = base × height, exactly as for the rectangle it becomes.

Which length is the height

The height is the straight-up distance between the two bases, not the slanted side. The slanted side is longer and gives too big an answer.

Cut and slide

Cut a right-angled triangle off one end of a parallelogram and slide it to the other end. You get a rectangle with the same base and height, and no area was added or lost. So the area is base × height.

Height is perpendicular, not slanted

As with triangles, the height is measured at a right angle to the base. The slanted side is longer than the height, and using it gives an answer that is too big. Diagrams usually mark the perpendicular height with a small square.

Many parallelograms, one area

Slant a parallelogram further while keeping its base and height fixed and the area is unchanged, even though the shape looks quite different and its perimeter grows. Area depends on base and height alone.

A rectangle is a parallelogram

A rectangle satisfies the definition of a parallelogram, so the formula base × height applies to it too — and reduces to length × width. One formula covering both is a sign the hierarchy of shapes is doing real work.