Rectangles with Same Area - Properties & Practice for 3rd Grade

Learn how different rectangles can have the same area through interactive exercises and examples.

Rectangles with Same Area Exercises
1 Finding rectangles with same area
Exercise 1
Rectangle A: 4 cm × 6 cm
Rectangle B: 3 cm × 8 cm
Do they have the same area?
Definition:

Same Area: Two shapes have the same area if they cover the same amount of space

Method for comparing areas:
  1. Calculate area of first rectangle: length × width
  2. Calculate area of second rectangle: length × width
  3. Compare the results
  4. If equal, they have the same area
Rectangle A
4 × 6 = 24
Rectangle B
3 × 8 = 24
Comparison
24 = 24
Step 1: Calculate area of Rectangle A

Area = length × width = 4 cm × 6 cm = 24 square cm

Step 2: Calculate area of Rectangle B

Area = length × width = 3 cm × 8 cm = 24 square cm

Step 3: Compare the areas

24 square cm = 24 square cm

Yes, both rectangles have the same area of 24 square cm
Final answer:

Yes, both rectangles have the same area

Applied rules:

• Area formula: Area = length × width

• Comparison: Equal products mean same area

• Multiplication: Calculate each rectangle's area separately

2 Creating rectangles with same area
Exercise 2
Find another rectangle that has the same area as a 5 cm × 4 cm rectangle
Definition:

Factor pairs: Numbers that multiply together to make the same product

Original
5 × 4 = 20
Find factors
20 = 2×10, 1×20, etc.
New rectangle
2 × 10
Step 1: Find the area of original rectangle

Area = 5 cm × 4 cm = 20 square cm

Step 2: Find factor pairs of 20

1 × 20 = 20, 2 × 10 = 20, 4 × 5 = 20

Step 3: Choose a new pair (not 5×4)

Let's use 2 × 10

Step 4: Verify the new rectangle

Area = 2 cm × 10 cm = 20 square cm

Rectangle 2 cm × 10 cm has same area as 5 cm × 4 cm
Final answer:

A rectangle with dimensions 2 cm × 10 cm has the same area

Applied rules:

• Factor pairs: Different pairs can multiply to same product

• Area conservation: Same product means same area

• Dimension flexibility: Same area can have different dimensions

3 Real-world application
Exercise 3
A garden plot is 6 ft × 4 ft. Another plot is 8 ft × 3 ft. Which has more space for planting?
Definition:

Space for planting: The area determines available growing space

Plot 1
6 × 4 = 24
Plot 2
8 × 3 = 24
Conclusion
Equal space
Step 1: Calculate area of first plot

Area = 6 ft × 4 ft = 24 square ft

Step 2: Calculate area of second plot

Area = 8 ft × 3 ft = 24 square ft

Step 3: Compare the areas

Both plots have 24 square feet of space

Both plots have the same space for planting: 24 square ft
Final answer:

Both plots have the same amount of space for planting

Applied rules:

• Real-world application: Area measures usable space

• Equality principle: Same area means same space

• Measurement: Units must match when comparing

Rectangles Comparison Visual
4 cm
6 cm
Area = 24 sq cm
8 cm
3 cm
Area = 24 sq cm
4 × 6 = 8 × 3 = 24
Different dimensions, same area
2 cm
10 cm
Area = 20 sq cm
4 cm
5 cm
Area = 20 sq cm
8 cm
2.5 cm
Area = 20 sq cm
Equal Products: Different factors can make same product
Different Dimensions: Same area can have various shapes
Conservation: Area stays constant despite shape changes
Summary & Q&A
Detailed Summary: Rectangles with Same Area
Definition: Two rectangles have the same area if they cover the same amount of space, regardless of their dimensions. This happens when the product of length and width is the same for both rectangles.
Core Rule: Area = length × width. If two rectangles have dimensions (l₁, w₁) and (l₂, w₂), they have the same area if l₁ × w₁ = l₂ × w₂.
Method: To determine if rectangles have the same area, calculate each area separately using the formula and compare the results.
Examples: - Rectangle 4×6 = 24 sq units and Rectangle 3×8 = 24 sq units (same area) - Rectangle 2×10 = 20 sq units and Rectangle 5×4 = 20 sq units (same area)
Tips: - Always multiply length by width to find area - Remember that different dimensions can yield the same area - Use factor pairs to find rectangles with the same area
Common Pitfall: Don't confuse area with perimeter. Rectangles with the same area can have different perimeters.
Memorization Aid: "Same product, same space" - if the multiplication gives the same result, the rectangles cover the same area.
Tip 1: Use factor pairs to find rectangles with the same area
Tip 2: Draw rectangles to visualize different dimensions with same area
Tip 3: Always label your units when calculating area
Tip 4: Remember: area measures space inside, perimeter measures around the edge

Q&A Section

Question: Can two rectangles have the same area but look completely different?

Answer: Absolutely yes! Two rectangles can have the same area but look very different:

  • A rectangle that is 1 cm × 12 cm (very long and thin) has an area of 12 square cm
  • A rectangle that is 3 cm × 4 cm (more square-like) also has an area of 12 square cm
  • Even a square that is 2√3 cm × 2√3 cm ≈ 3.46 cm × 3.46 cm has an area of 12 square cm

They all cover the same amount of space (12 square cm) but have very different shapes. This is why we say "same area, different dimensions."

Question: How do I find other rectangles that have the same area as one I already have?

Answer: You need to find factor pairs of the area! Here's how:

  • If you have a rectangle with area 12 square units (like 3 × 4), find all number pairs that multiply to 12
  • Factor pairs of 12: 1×12, 2×6, 3×4
  • This means rectangles with dimensions 1×12, 2×6, and 3×4 all have the same area of 12 square units

To find factor pairs systematically, divide the area by 1, then 2, then 3, etc., until you start repeating pairs. Any division that results in a whole number gives you a valid rectangle dimension pair.

Question: Do rectangles with the same area also have the same perimeter?

Answer: No, rectangles with the same area usually have different perimeters! Here's an example:

  • Rectangle 1: 4 × 6 → Area = 24 sq units, Perimeter = 2(4+6) = 20 units
  • Rectangle 2: 3 × 8 → Area = 24 sq units, Perimeter = 2(3+8) = 22 units
  • Rectangle 3: 2 × 12 → Area = 24 sq units, Perimeter = 2(2+12) = 28 units

Notice how all three rectangles have the same area (24 sq units) but different perimeters (20, 22, and 28 units). This shows that area and perimeter are independent properties of rectangles.

The square (closest to a square) typically has the smallest perimeter for a given area.