3B · Chapter 13 · Lesson 7
Area and Perimeter
Compare configurations with equal area but different perimeters, and vice versa.
Today's mathematical question
A 6-by-5 rectangle has what area and what perimeter? Include units.
Learning target
Compare configurations with equal area but different perimeters, and vice versa.
Before starting: Prior lesson: Perimeter of Rectangles. Try the warm-up independently.
Prerequisite · 3 questions
Check earlier skills first
These are earlier ideas you need for today; revisit them if they are difficult.
- P1
A rectangle measures 4 cm by 2 cm. What is its perimeter?
Reveal explanation
Answer: 12 cm
Walk along every outer edge once: 4 + 2 + 4 + 2 = 12 cm.
Square-unit grid; count only visible squares without gaps or overlap. - P2
A rectangle measures 3 cm by 2 cm. What is its perimeter?
Reveal explanation
Answer: 10 cm
Walk along every outer edge once: 3 + 2 + 3 + 2 = 10 cm.
Square-unit grid; count only visible squares without gaps or overlap. - P3
A rectangle measures 7 cm by 2 cm. What is its perimeter?
Reveal explanation
Answer: 18 cm
Walk along every outer edge once: 7 + 2 + 7 + 2 = 18 cm.
Square-unit grid; count only visible squares without gaps or overlap.
Learn and understand
Area and perimeter describe different attributes of the same shape. Area counts squares inside; perimeter counts units around the boundary. Two rectangles may each cover 12 squares but have different perimeters: 3 × 4 has perimeter 14 and 2 × 6 has perimeter 16. Always include correct units. A 3-by-4 rectangle has area 12 square units and perimeter 14 units. It is possible for two shapes to have the same area but different perimeters.
Represent it three ways
- Concrete: Build equal pieces, measures, counters or shapes with real objects when useful.
- Pictorial: Draw two different rectangles on squared paper with equal area; trace their boundaries in different colors and compare.
- Abstract: Write the matching number sentence, calculation, comparison or mathematical description. Explain each number's unit.
Important vocabulary
- area
- number of interior square units
- perimeter
- distance around boundary
- dimension
- length of a side
Fully worked examples and checks
Try each new understanding check before revealing the explanation.
Worked example 1
Explore, calculate, verify
A 6-by-5 rectangle has what area and what perimeter? Include units.
- Identify the whole, the units, and what the question asks.
- Area counts square units: 6 × 5 = 30. Perimeter counts the boundary: 2(6+5) = 22. They have different units.
- Check that the answer uses the right units and makes sense.
Answer: 30 square units; 22 units
Check your understanding 1
A 3-by-4 rectangle has what area and what perimeter? Include units.
Reveal answer and explanation
12 square units; 14 units — Area counts square units: 3 × 4 = 12. Perimeter counts the boundary: 2(3+4) = 14. They have different units.
Worked example 2
Explore, calculate, verify
A 4-by-3 rectangle has what area and what perimeter? Include units.
- Identify the whole, the units, and what the question asks.
- Area counts square units: 4 × 3 = 12. Perimeter counts the boundary: 2(4+3) = 14. They have different units.
- Check that the answer uses the right units and makes sense.
Answer: 12 square units; 14 units
Check your understanding 2
A 8-by-4 rectangle has what area and what perimeter? Include units.
Reveal answer and explanation
32 square units; 24 units — Area counts square units: 8 × 4 = 32. Perimeter counts the boundary: 2(8+4) = 24. They have different units.
Worked example 3
Explore, calculate, verify
A 8-by-3 rectangle has what area and what perimeter? Include units.
- Identify the whole, the units, and what the question asks.
- Area counts square units: 8 × 3 = 24. Perimeter counts the boundary: 2(8+3) = 22. They have different units.
- Check that the answer uses the right units and makes sense.
Answer: 24 square units; 22 units
Check your understanding 3
A 6-by-6 rectangle has what area and what perimeter? Include units.
Reveal answer and explanation
36 square units; 24 units — Area counts square units: 6 × 6 = 36. Perimeter counts the boundary: 2(6+6) = 24. They have different units.
Guided · 5 questions
Guided practice
Keep the model and earlier examples visible while practising.
- G1
A 5-by-5 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 25 square units; 20 units
Area counts square units: 5 × 5 = 25. Perimeter counts the boundary: 2(5+5) = 20. They have different units.
Square-unit grid; count only visible squares without gaps or overlap. - G2
A 5-by-6 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 30 square units; 22 units
Area counts square units: 5 × 6 = 30. Perimeter counts the boundary: 2(5+6) = 22. They have different units.
Square-unit grid; count only visible squares without gaps or overlap. - G3
A 9-by-4 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 36 square units; 26 units
Area counts square units: 9 × 4 = 36. Perimeter counts the boundary: 2(9+4) = 26. They have different units.
Square-unit grid; count only visible squares without gaps or overlap. - G4
A 5-by-3 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 15 square units; 16 units
Area counts square units: 5 × 3 = 15. Perimeter counts the boundary: 2(5+3) = 16. They have different units.
Square-unit grid; count only visible squares without gaps or overlap. - G5
A 4-by-6 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 24 square units; 20 units
Area counts square units: 4 × 6 = 24. Perimeter counts the boundary: 2(4+6) = 20. They have different units.
Square-unit grid; count only visible squares without gaps or overlap.
Independent · 8 questions
Independent practice
Work these without hints and explain the method used. Write answers on paper.
- I1
A 9-by-3 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 27 square units; 24 units
Area counts square units: 9 × 3 = 27. Perimeter counts the boundary: 2(9+3) = 24. They have different units.
Square-unit grid; count only visible squares without gaps or overlap. - I2
A 5-by-4 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 20 square units; 18 units
Area counts square units: 5 × 4 = 20. Perimeter counts the boundary: 2(5+4) = 18. They have different units.
Square-unit grid; count only visible squares without gaps or overlap. - I3
A 6-by-4 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 24 square units; 20 units
Area counts square units: 6 × 4 = 24. Perimeter counts the boundary: 2(6+4) = 20. They have different units.
Square-unit grid; count only visible squares without gaps or overlap. - I4
A 6-by-3 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 18 square units; 18 units
Area counts square units: 6 × 3 = 18. Perimeter counts the boundary: 2(6+3) = 18. They have different units.
Square-unit grid; count only visible squares without gaps or overlap. - I5
A 3-by-2 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 6 square units; 10 units
Area counts square units: 3 × 2 = 6. Perimeter counts the boundary: 2(3+2) = 10. They have different units.
Square-unit grid; count only visible squares without gaps or overlap. - I6
A 9-by-5 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 45 square units; 28 units
Area counts square units: 9 × 5 = 45. Perimeter counts the boundary: 2(9+5) = 28. They have different units.
Square-unit grid; count only visible squares without gaps or overlap. - I7
A 4-by-2 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 8 square units; 12 units
Area counts square units: 4 × 2 = 8. Perimeter counts the boundary: 2(4+2) = 12. They have different units.
Square-unit grid; count only visible squares without gaps or overlap. - I8
A 9-by-6 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 54 square units; 30 units
Area counts square units: 9 × 6 = 54. Perimeter counts the boundary: 2(9+6) = 30. They have different units.
Square-unit grid; count only visible squares without gaps or overlap.
Challenge · 2 questions
Challenge and error analysis
Go beyond arithmetic: justify a claim, correct reasoning, or connect two representations.
- C1
A 3-by-3 rectangle and a 1-by-9 rectangle have the same area. Must they have the same perimeter?
Reveal explanation
Answer: No
Both areas are equal, but different side dimensions usually give different boundary lengths.
- C2
A student says a rectangle with area 9 square units must have perimeter 9 units. Explain why this is not generally true using a 3-by-3 example.
Reveal explanation
Answer: Perimeter 12 units
The area is 3 × 3 = 9 square units, but its perimeter is 2 × (3+3) = 12 units. Area and perimeter measure different things.
Exit · 3 questions
Exit check
Solve independently, then return on another day to check retention.
- E1
A 3-by-3 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 9 square units; 12 units
Area counts square units: 3 × 3 = 9. Perimeter counts the boundary: 2(3+3) = 12. They have different units.
Square-unit grid; count only visible squares without gaps or overlap. - E2
A 8-by-6 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 48 square units; 28 units
Area counts square units: 8 × 6 = 48. Perimeter counts the boundary: 2(8+6) = 28. They have different units.
Square-unit grid; count only visible squares without gaps or overlap. - E3
A 8-by-2 rectangle has what area and what perimeter? Include units.
Reveal explanation
Answer: 16 square units; 20 units
Area counts square units: 8 × 2 = 16. Perimeter counts the boundary: 2(8+2) = 20. They have different units.
Square-unit grid; count only visible squares without gaps or overlap.