Comparing Rectangles with Same Perimeter, Different Area (and Vice Versa) - Third Grade Mathematics

One of the most fascinating discoveries in geometry is that two shapes can have the exact same perimeter while having completely different areas, or they can have the exact same area while having completely different perimeters! Perimeter and area are independent geometric properties. In this chapter, you will investigate the deep relationship between perimeter and area and discover how the shape of a rectangle affects the surface space inside.

The Big Idea: Perimeter vs. Area

Let us refresh the core definitions:
- Perimeter: The distance around the boundary (fence).

- Area: The surface space covered inside (grass).


Just because two backyards require the exact same length of fencing does NOT mean they have the same amount of grass to play on!

Case 1: Same Perimeter, Different Areas

Suppose you have 20 feet of fencing. You can build several different rectangles that all have a perimeter of 20 feet:

Let us test different rectangles with Perimeter = 20 feet:

Rectangle A (Long and skinny):
Length = 9 ft, Width = 1 ft
Perimeter = 9 + 1 + 9 + 1 = 20 feet
Area = 9 ft x 1 ft = 9 square feet

Rectangle B (Medium):
Length = 8 ft, Width = 2 ft
Perimeter = 8 + 2 + 8 + 2 = 20 feet
Area = 8 ft x 2 ft = 16 square feet

Rectangle C (More balanced):
Length = 7 ft, Width = 3 ft
Perimeter = 7 + 3 + 7 + 3 = 20 feet
Area = 7 ft x 3 ft = 21 square feet

Rectangle D (Square):
Length = 5 ft, Width = 5 ft
Perimeter = 5 + 5 + 5 + 5 = 20 feet
Area = 5 ft x 5 ft = 25 square feet!

Look at the comparison table:

+-------------+------------------+-------------------+-------------------+
| Rectangle   | Dimensions       | Perimeter         | Area              |
+-------------+------------------+-------------------+-------------------+
| A           | 9 ft by 1 ft     | 20 ft             | 9 sq ft           |
| B           | 8 ft by 2 ft     | 20 ft             | 16 sq ft          |
| C           | 7 ft by 3 ft     | 20 ft             | 21 sq ft          |
| D (Square)  | 5 ft by 5 ft     | 20 ft             | 25 sq ft (MAX!)   |
+-------------+------------------+-------------------+-------------------+

The Maximization Rule:

For a fixed perimeter, the rectangle that is closest to a square (where length and width are as equal as possible) always encloses the GREATEST area! Long, skinny rectangles have smaller areas.

Case 2: Same Area, Different Perimeters

Now let us look at the reverse situation. Suppose you need to create a garden with an area of exactly 12 square meters:

Let us find all whole-number rectangles with Area = 12 square meters:

Rectangle 1 (Long and narrow):
1 m by 12 m
Area = 1 x 12 = 12 sq meters
Perimeter = 1 + 12 + 1 + 12 = 26 meters!

Rectangle 2 (Medium):
2 m by 6 m
Area = 2 x 6 = 12 sq meters
Perimeter = 2 + 6 + 2 + 6 = 16 meters

Rectangle 3 (Most square-like):
3 m by 4 m
Area = 3 x 4 = 12 sq meters
Perimeter = 3 + 4 + 3 + 4 = 14 meters!

Look at the comparison table:

+-------------+------------------+-------------------+-------------------+
| Rectangle   | Dimensions       | Area              | Perimeter         |
+-------------+------------------+-------------------+-------------------+
| 1           | 1 m by 12 m      | 12 sq m           | 26 m (Largest!)   |
| 2           | 2 m by 6 m       | 12 sq m           | 16 m              |
| 3           | 3 m by 4 m       | 12 sq m           | 14 m (Smallest!)  |
+-------------+------------------+-------------------+-------------------+

The Minimization Rule:

For a fixed area, the rectangle that is closest to a square requires the LEAST amount of perimeter! Long, skinny shapes stretch out the boundary, requiring much more perimeter to enclose the same area.

Chapter Practice Exercises

  1. Two rectangles both have a perimeter of 16 cm: Rectangle A is 7 cm by 1 cm. Rectangle B is 5 cm by 3 cm. a. Calculate the area of Rectangle A. b. Calculate the area of Rectangle B. c. Which rectangle has the greater area?
  2. Two rectangles both have an area of 24 square inches: Rectangle X is 24 inches by 1 inch. Rectangle Y is 6 inches by 4 inches. a. Calculate the perimeter of Rectangle X. b. Calculate the perimeter of Rectangle Y. c. Which rectangle has the smaller perimeter?
  3. A farmer has 24 meters of fence to build a pen for his sheep. What dimensions (length and width) should he choose to give the sheep the greatest possible grassy area? What is that area?
  4. Can two rectangles have the exact same perimeter and the exact same area, but different side lengths? (Try comparing a 2 by 6 rectangle with other whole-number rectangles).
  5. A designer is designing a dog run with an area of 36 square feet. a. Name two different rectangular dimensions that give this area. b. Which design requires less fencing?

Solutions and Step-by-Step Answers

  1. Perimeters of 16 cm: a. Area A = 7 cm x 1 cm = 7 square cm. b. Area B = 5 cm x 3 cm = 15 square cm. c. Rectangle B has a much greater area (15 sq cm > 7 sq cm).
  2. Areas of 24 square inches: a. Perimeter X = 24 + 1 + 24 + 1 = 50 inches. b. Perimeter Y = 6 + 4 + 6 + 4 = 20 inches. c. Rectangle Y has the smaller perimeter (20 in < 50 in).
  3. Greatest area for 24 meters of fence: The shape closest to a square provides maximum area. Since 24 ÷ 4 = 6, a 6 meter by 6 meter square pen gives the greatest area: Area = 6 m x 6 m = 36 square meters.
  4. For whole-number rectangles, if both the perimeter and area are identical, the dimensions must be identical. For example, a 2 by 6 rectangle has Perimeter = 16 and Area = 12; no other whole-number rectangle has both P = 16 and A = 12.
  5. Dog run with area 36 sq ft: a. Dimensions could be 4 ft by 9 ft, or 6 ft by 6 ft (or 3 ft by 12 ft). b. The 6 ft by 6 ft square requires less fencing (Perimeter = 24 ft) compared to the 4 ft by 9 ft rectangle (Perimeter = 26 ft).