Cut off the slanted end
A parallelogram is a four-sided shape whose opposite sides are parallel, like a rectangle that has been pushed over. This one has a base of 7. Its height is 4: the distance straight up from the base to the top side, measured at a right angle to the base.
Cut along the dashed line, which runs straight down from a top corner to the base. The cut takes a right-angled triangle off the slanted end.
The dashed line runs straight down from the top corner and cuts off the slanted end.
Slide it to the other end
Slide the triangle along to the other end of the shape. Its slanted side fits exactly against the other slanted side, because opposite sides of a parallelogram are parallel and the same length. It leaves a gap at one end and fills the matching gap at the other.
No piece was thrown away and none was added, so the area is the same as before.
The triangle leaves the left end and fills the gap at the right end: the same two pieces, so the same area.
Now it is a rectangle
The two pieces now make a rectangle 7 long and 4 high. It holds 4 rows of 7 squares, so its area is 7 × 4 = 28, and the parallelogram’s area is 28 as well.
The rectangle’s length is the parallelogram’s base, and the rectangle’s height is the parallelogram’s height. The cut and the slide did not change either one.
4 rows of 7 squares: the rectangle’s area is 7 × 4 = 28.
Base times height
The same cut and slide turn every parallelogram into a rectangle with the same base and the same height. So the area of a parallelogram is its base times its height: area of a parallelogram = b × h.
A parallelogram with a base of 9 cm and a height of 5 cm has an area of 9 × 5 = 45 cm².
Area of a parallelogram = b × h, where the height h is measured at a right angle to the base b.
Leaning further
Push the top side farther along and the parallelogram leans more. Its sloping sides get longer, but the top side stays on the same line, so the height does not change, and neither does the base. The area stays base × height.
This is why the length of a sloping side is never used for the area. It is longer than the height, and multiplying by it would give too much.
6 × 4 = 24
Try to change the area
Drag the top corner to lean the rectangle over. The wedge cut from one end is the same as the wedge added at the other, so the area stays 6 × 4 = 24.
Not half, and not the perimeter
Two mistakes are common. Halving, 7 × 4 ÷ 2 = 14, is the rule for a triangle; a parallelogram becomes a whole rectangle, so nothing is halved. And 2 × (7 + 4) = 22 is the perimeter of a 7 by 4 rectangle, the distance around its edge, not the space inside it.