The Rectangular Garden Fence
A class plans a rectangular garden with an area of 36 square metres. The fence comes in 1-metre panels, and the school will pay for the perimeter only. Group A proposes a 6 m by 6 m garden. Group B proposes a 4 m by 9 m garden. Group C proposes a 1 m by 36 m garden. All three gardens have the same area, but the groups disagree about which needs the least fencing. The teacher also reminds them that the panels must meet at right angles and that the garden must have whole-metre side lengths.
1. Calculate the perimeter of each proposed garden and determine which design uses the least fencing. Explain why equal areas can have different perimeters.
Read the worked answer
Perimeter of A: 2 × (6 + 6) = 24 m. Perimeter of B: 2 × (4 + 9) = 26 m. Perimeter of C: 2 × (1 + 36) = 74 m. Group A's 6 m by 6 m square uses the least fencing, 24 m. All three have area 36 m² because length × width = 36 in each case, but perimeter depends on how the area is arranged. A shape closer to a square packs the same area into a more compact boundary; a long thin rectangle spreads the area out and needs more fence. So area and perimeter measure different things and cannot be inferred from one another.
2. The school budget allows at most 26 m of fencing. List which of the three designs are affordable, and explain whether a garden with area 36 m² could ever need less than 24 m of fence using whole-metre sides.
Read the worked answer
Groups A (24 m) and B (26 m) are within budget; Group C (74 m) is not. To test whether less than 24 m is possible, list whole-number factor pairs of 36: 1×36 (74 m), 2×18 (40 m), 3×12 (30 m), 4×9 (26 m), 6×6 (24 m). The smallest perimeter is 24 m at 6×6, so no whole-metre rectangle with area 36 m² can need less than 24 m. The square is the most compact option. This confirms that 24 m is the minimum, not just the best of the three guesses.