the area of a parallelogram
Take a rectangle drawn on paper, slice a right triangle off one end, and slide it round to the other end. Nothing is added or removed, so the area is unchanged — but the shape is now a slanted parallelogram. This cut-and-slide trick is the whole story of why a parallelogram's area looks just like a rectangle's.
The formula is A = b · h, where b is the length of one side chosen as the base and h is the perpendicular height — the straight-up distance between that base and the opposite side, measured at a right angle, NOT the length of the slanted side. So if a parallelogram has a base of 6 and a perpendicular height of 4, its area is 24, regardless of how far it leans.
The single most common mistake is to multiply the base by the slanted side instead of by the perpendicular height. The slanted side is always at least as long as the height, so using it overstates the area. When the parallelogram leans a lot, the height can be much shorter than the side — that gap is exactly where the cut-and-slide intuition keeps you honest.
A parallelogram has a base of 10 cm and slanted sides of 6 cm, but its perpendicular height is only 5 cm. Its area is 10 × 5 = 50 cm^2 — the 6 cm side never enters the calculation.
Use the perpendicular height, never the slanted side.
The height must be measured perpendicular to the chosen base; the slanted side overstates the area unless the parallelogram happens to be a rectangle.