Area of a parallelogram

A parallelogram has no unit squares lining up neatly inside it. So we turn it into a rectangle, without adding or losing any area.

In parallelogram ABCD, draw AX perpendicular to CD. That perpendicular is called a height of the parallelogram. It cuts the shape into ∆AXD and the trapezium ABCX.

Now slide ∆AXD across to the other end. Does it fit exactly? Check properly, not just by eye.

Angle X is 90°, and since AB ∥ XC, angle A is 90° too. Extend XC to the right and drop a perpendicular from B to meet it at Y. The triangle ∆BYC is exactly what ABCX needs to become a rectangle.

Is ∆AXD congruent to ∆BYC?

• BY = AX, since ABYX is a rectangle.
• ∠BYC = ∠AXD = 90°.
• BC = AD, because opposite sides of a parallelogram are equal.

By the RHS congruence rule, the triangles match, so ∆AXD fits exactly into the gap.

Cutting a shape into pieces and rearranging them into a different shape of the same area is called dissection. It is the method behind every formula in the rest of this chapter.

So Area(ABCD) = Area(rectangle ABYX) = AX × XY. AX is the height. And XY equals DC, because DX = CY, and adding the shared piece XC to each gives DC = XY. DC is the base.

Area of a parallelogram = base × height. Any side can be the base: cut along a perpendicular to a different side and the same rearrangement works, as long as you use that side's own height.