Questions: Relationship Between Area and Perimeter

Question 1

Rectangle A is 1 unit × 12 units. Rectangle B is 3 units × 4 units. What is true about their area and perimeter?

• A. They have the same area and the same perimeter
• B. They have the same area (12 square units) but different perimeters (26 vs. 14)
• C. They have different areas but the same perimeter
• D. A larger perimeter always means a larger area

Answer: (B)

Both rectangles have area = 12 square units (1×12 and 3×4). But their perimeters differ: Rectangle A has perimeter 2(1+12) = 26; Rectangle B has perimeter 2(3+4) = 14. This demonstrates that area and perimeter are independent — knowing one tells you nothing predictable about the other. Option D is the most common misconception.

Question 2

A farmer needs to buy fencing for a rectangular field and also sod to cover the ground inside. Which measurement does each purchase require?

• A. Fencing requires area; sod requires perimeter
• B. Fencing requires perimeter; sod requires area
• C. Both fencing and sod require area
• D. Both fencing and sod require perimeter

Answer: (B)

Fencing goes around the boundary of the field, so it depends on perimeter (the total length of all sides). Sod covers the interior surface, so it depends on area (the amount of space enclosed). This is the real-world meaning of the distinction: perimeter is for borders and boundaries; area is for surfaces and interiors. Confusing them leads to buying far too much or too little material.

Question 3

Two rectangles can have the same area but different perimeters.

• T. True
• F. False

Answer: True

This is the central insight of this topic. A 1×12 rectangle and a 3×4 rectangle both have area 12, but their perimeters are 26 and 14 respectively. Area and perimeter are independent properties — changing the shape of a figure (while keeping area constant) changes its perimeter, and vice versa.

Question 4

If you increase the perimeter of a rectangle, its area should also increase.

• T. True
• F. False

Answer: False

Area and perimeter are independent — one can change while the other stays the same or even moves in the opposite direction. For example, a 1×7 rectangle has perimeter 16 and area 7; a 3×5 rectangle also has perimeter 16 but area 15. Or a 2×8 rectangle has perimeter 20 and area 16, while a 4×4 has perimeter 16 and area 16 — smaller perimeter, same area. There is no rule that links them in a predictable direction.

Question 5

Explain why area and perimeter are independent measurements, and give an example showing they can differ for same-area shapes.

Model answer: Area measures the interior space (length × width for rectangles); perimeter measures the total boundary length (sum of all sides). They measure different things, so they don't move together. Example: a 2×6 rectangle and a 3×4 rectangle both have area 12 square units, but perimeters of 16 and 14 respectively.

The independence comes from the fact that the same amount of interior space can be arranged into different shapes — long and thin, or short and wide — with very different boundary lengths. This is why real-world problems must specify which measurement they need: fencing (perimeter) and carpeting (area) are genuinely different questions even about the same physical space.