You know the area of a rectangle is length times width, even with fraction side lengths. A parallelogram leans over, but you can turn it into a rectangle without changing its area.
Key idea
The area of a parallelogram is base times height: $A = bh$. The height is the distance straight across from the base to the opposite side, measured at a right angle to the base, not along the slanted side.
Why it works
Cut along the height to take a right triangle off one end, and slide it to the other end. The pieces now form a rectangle. Nothing was added or lost, so the area is the same. The rectangle's length is the parallelogram's base, and its width is the parallelogram's height, so the area is $b \times h$.
Worked example: Base and height
A parallelogram has base 9 cm and height 5 cm. Find its area.
- $A = bh = 9 \times 5$.
Answer: 45 square centimeters
Worked example: Ignore the slanted side
A parallelogram has a base of 7.5 cm, slanted sides of 5 cm, and a height of 4 cm. Find its area.
- The height is 4 cm; the 5 cm side leans, so it is not the height.
- $A = 7.5 \times 4 = 30$.
Answer: 30 square centimeters
Worked example: Work backward
A parallelogram has an area of 54 square inches and a base of 9 inches. What is its height?
- $54 = 9 \times h$.
- $h = 54 \div 9 = 6$.
Answer: 6 inches
Common mistake
Multiplying the base by the slanted side. In the second example that gives $7.5 \times 5 = 37.5$, which is too big. Quick check: push a parallelogram over until it is almost flat. Its sides keep their lengths, but its area shrinks toward 0, so the slanted side cannot be what decides the area. Only the height can.