In Grade 3 you found area and perimeter by counting squares and adding sides. With two-digit multiplication you can use formulas for rectangles of any size.
Key idea
For a rectangle with length $l$ and width $w$: area $A = l \times w$ (square units) and perimeter $P = 2 \times l + 2 \times w$ (units). To find a missing side, work backward: $w = A \div l$, or take $2 \times l$ away from the perimeter and halve what is left.
Why it works
Area counts unit squares: $w$ rows of $l$ squares is $l \times w$. Perimeter walks around the outside: along a length, a width, the other length and the other width. Opposite sides of a rectangle are equal, so that is two lengths and two widths.
Worked example: Area and perimeter
A garden is 12 feet long and 7 feet wide. Find its area and perimeter.
- Area: $12 \times 7 = 84$ square feet.
- Perimeter: $2 \times 12 + 2 \times 7 = 24 + 14 = 38$ feet.
Answer: Area 84 sq ft; perimeter 38 ft
Worked example: A missing side from the perimeter
A rectangle has a perimeter of 30 cm and a length of 9 cm. How wide is it?
- The two lengths use $2 \times 9 = 18$ cm.
- The two widths share the rest: $30 - 18 = 12$ cm.
- One width: $12 \div 2 = 6$ cm. Check: $9 + 6 + 9 + 6 = 30$.
Answer: 6 cm
Worked example: A missing side from the area
A rug has an area of 96 square feet and a length of 12 feet. How wide is it?
- $12 \times w = 96$.
- $w = 96 \div 12 = 8$.
Answer: 8 feet
Common mistake
Adding the length and width only once for the perimeter ($12 + 7 = 19$ feet). That is only half the trip around. Trace all four sides with your finger, or use $2 \times l + 2 \times w$.