Area of a parallelogram

The idea

Area of a parallelogram = base × height, the height being perpendicular to the base; moving a cut-off right triangle dissects it into a rectangle, and any side with its corresponding height can be used.

Leaning shape

A parallelogram leans to the right. The dashed rectangle has the same base and the same height as the parallelogram: both are 7 wide at the bottom and 4 high.

Which has the greater area?

• The rectangle
• The parallelogram, because it leans out further
• Both have the same area

Cut along the height

The dashed line from the top corner straight down is the height. The parallelogram is cut along it. Slide the small triangle to the other end to make the dashed rectangle, then look at both sides as the base in the second tab.

Base times height

Area of a parallelogram = base × height, the height being perpendicular to the base. Moving the cut-off right triangle dissects it into a rectangle, and any side with its corresponding height can be used.

The height meets the base at a right angle. Cut along it and slide the small triangle to the other end: the parallelogram becomes a rectangle 7 long and 4 high, so the area is 7 × 4 = 28. The length of the rectangle is the base and its width is the height.

The moved triangle fits exactly: it and the gap at the other end are right triangles with the same height and with equal slanted sides (the opposite sides of a parallelogram are equal), so they are the same size.

Any side can be the base, with its own height. The second tab shows base 10 with height 4, and base 5 with height 8: 10 × 4 = 40 and 5 × 8 = 40. The height is always measured at a right angle, never along the slanted side.

Notes

Area of a parallelogram = base × height, the height being perpendicular to the base. Moving the cut-off right triangle dissects it into a rectangle, and any side with its corresponding height can be used.

Check yourself

Parallelograms P and Q both have a base of 8 cm and a height of 5 cm. P is almost upright. Q leans a lot, so its slanted sides are much longer. Which has the greater area?

A parallelogram has a base of 12 cm, a height of 5 cm and a slanted side of 6 cm. What is its area, in cm²?

Answer: 60

Area = base × height = 12 × 5 = 60 cm². The slanted side is not needed.

This parallelogram has sides 10 and 5. The height on the side 10 is 4. What is the height on the side 5, in units?

Answer: 8

Area = 10 × 4 = 40. Using the side 5 as the base: 5 × height = 40, so the height is 8.

Which pair gives the area of a parallelogram?

• Q, its slanted sides are longer. The slanted sides do not matter. The area is base × height = 8 × 5 = 40 cm² for both.
• P, it stands more upright. Leaning does not change the height. Both have height 5 cm, so both have area 8 × 5 = 40 cm².
• Both have the same area — correct. Yes! Same base and same height give 8 × 5 = 40 cm², however much the parallelogram leans.
• Two neighbouring sides, multiplied. Two neighbouring sides do not meet at a right angle in a leaning parallelogram, so their product is bigger than the area.
• A side and the height perpendicular to that side, multiplied — correct. Yes! Base × height, with the height measured at a right angle to the chosen base.
• The two slanted sides, added. Adding lengths gives a perimeter-like number, not an area.
• A side and half of the height. There is no half in the parallelogram: it becomes a whole rectangle of base × height.