A farmer has 24 meters of fencing and wants to make the biggest rectangular garden possible. A neighbor says, 'It doesn't matter how you shape it — all rectangles with the same perimeter have the same area.' Is the neighbor right?
AYes — same perimeter always means same area for rectangles
BNo — a 1×11 rectangle and a 6×6 rectangle both have perimeter 24, but areas of 11 and 36 square meters respectively
CNo — but only if one rectangle is a square and the other isn't
DYes — area and perimeter are always equal for the same shape
This is the central misconception this topic addresses: perimeter and area are independent. With 24 meters of fencing you could make a 1×11 rectangle (area = 11 m²), a 4×8 rectangle (area = 32 m²), or a 6×6 square (area = 36 m²) — all with perimeter 24, but wildly different areas. The neighbor is wrong, and recognizing this independence is the key insight.
Question 2 Multiple Choice
You are buying baseboard trim to run along the bottom of the walls of a rectangular room that is 5 meters long and 4 meters wide. Which calculation gives you the right amount to buy?
A5 × 4 = 20 square meters — you need to cover the floor surface
B(5 + 4) × 2 = 18 meters — you need the total length around the walls
C5 + 4 = 9 meters — you need the length and width added once
D5 × 4 × 2 = 40 — you need double the area
Baseboard trim runs along the edge of the room, so you need the perimeter — the total distance around all four walls. Area (length × width) measures how much surface is covered, which is what you'd need for flooring or carpet. Asking 'edge or surface?' is the fastest way to choose: trim is an edge problem, so perimeter is correct.
Question 3 True / False
If two rectangles have the same area, they is expected to also have the same perimeter.
TTrue
FFalse
Answer: False
A 1×12 rectangle and a 3×4 rectangle both have area 12 square units, but their perimeters are 26 and 14 units respectively — very different. Area and perimeter are independent measurements. A change in one does not determine the other, which is why you always need to read a problem carefully to know which one is being asked for.
Question 4 True / False
Area is measured in square units (like cm²) while perimeter is measured in plain length units (like cm).
TTrue
FFalse
Answer: True
Area counts the square tiles that fill a surface — each tile has two dimensions, so the unit is squared (cm², m², ft²). Perimeter measures a single length — the distance you would walk all the way around the outside — so it uses plain length units. If your answer is in square units when the question asks about fencing, you have calculated the wrong thing.
Question 5 Short Answer
Explain why two different rectangles can have the same perimeter but different areas. Use a specific numerical example to support your answer.
Think about your answer, then reveal below.
Model answer: Perimeter and area measure different things — perimeter is the total length around the outside edge, while area is the amount of surface inside. You can rearrange the same boundary length into different shapes, changing how much space is enclosed. For example, both a 2×8 rectangle and a 5×5 square have perimeter 20 units, but their areas are 16 and 25 square units respectively. The boundary stayed the same; the interior space changed.
The independence of area and perimeter surprises most students because they assume 'bigger boundary = more space inside.' But a long, thin rectangle can have a huge perimeter while enclosing very little area. This insight is practically important: a farmer with a fixed amount of fencing wants the shape that maximizes enclosed area, which requires understanding that the two measures are not locked together.