Equivalent figures and the area of a parallelogram — 11–13
Same area, different shape: cut and move pieces to see why a parallelogram has area base × height.
Same area, different shapes
Two figures are equivalent when they have the same area, even if their shapes differ. A parallelogram has opposite sides parallel. Its height is the distance between two parallel sides, measured at right angles, not the slanted side. Each base has its own height. A parallelogram is equivalent to the rectangle with the same base and the same height, so its area is base × height.
Why the area is base × height
Cut a right-angled triangle off one end of the parallelogram and stick it on the other end: you get a rectangle. Nothing was added or removed, only moved, so the area is the same. That rectangle has the same base and the same height as the parallelogram, and its area is base × height. If you slide the top side along its parallel line without changing the base or the height, the area does not change either.
Working out the area of a parallelogram
A parallelogram has a base of 8 cm, a slanted side of 5 cm and a height of 4 cm. Area = base × height = 8 × 4 = 32 cm². The 5 cm side is not used. If you take the 5 cm side as the base, the matching height changes: 5 × height = 32, so height = 32 ÷ 5 = 6.4 cm. Another equivalent rectangle is 16 cm by 2 cm, since 16 × 2 = 32. Same area, different shapes.
The trap: using the slanted side as the height
It is natural to multiply the two sides you can see: 8 × 5 = 40 cm². That is wrong, because the slanted side is not perpendicular to the base. Only the height, measured at right angles, gives the right area: 8 × 4 = 32 cm². Another mix-up: thinking equivalent figures must have the same perimeter. The parallelogram with sides 8 cm and 5 cm has perimeter 26 cm; the 8 cm by 4 cm rectangle has 24 cm, yet both have area 32 cm².
Where you see this outside school
A tile shaped like a parallelogram covers the same area as a rectangular tile with the same base and height, so the number of tiles to cover a floor is worked out from the area. You can also compare shapes on paper by cutting and moving pieces: if the pieces are the same, the area is the same, however much the shape changes.
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