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Perimeter vs Area: The Mistake That Changes the Whole Answer
Learn how perimeter measures the distance around a shape while area measures the space inside it, with worked examples, common traps and 36 original Grade 4 practice questions.
The idea
Perimeter is the total distance around a shape. Imagine walking along every outside edge once.
Area is the amount of flat space inside a shape. Imagine covering the inside with square tiles.
For a rectangle with length L and width W, perimeter is 2 × (L + W). Area is L × W. The formulas look different because they answer different questions.
Worked examples
Example 1 · Same rectangle, two different questions
A rectangle is 8 cm long and 3 cm wide. Find its perimeter and area.
- Perimeter: add all four sides: 8 + 3 + 8 + 3 = 22.
- Area: count the square units in 8 rows of 3, or multiply 8 × 3 = 24.
Perimeter = 22 cm; area = 24 cm².
Example 2 · A square
A square has side length 6 m. Find its perimeter and area.
- All four sides are 6 m, so perimeter = 4 × 6 = 24 m.
- Area = 6 × 6 = 36 m².
Perimeter = 24 m; area = 36 m².
Example 3 · Read the situation before calculating
A 12 m by 5 m garden needs a fence around it and soil to cover its surface. What does each job use?
- Fence goes around the outside, so use perimeter: 2 × (12 + 5) = 34 m.
- Soil covers the inside, so use area: 12 × 5 = 60 m².
34 m of fence and 60 m² of surface area.
Example 4 · A tempting wrong answer
A student says the perimeter of a 9 cm by 4 cm rectangle is 36 cm because 9 × 4 = 36. What went wrong?
- The multiplication 9 × 4 finds area, not the distance around the edge.
- Perimeter = 9 + 4 + 9 + 4 = 26 cm.
36 is the area in cm². The perimeter is 26 cm.
Example 5 · Work backwards
A rectangle has perimeter 30 cm and length 9 cm. Find its width and area.
- Half the perimeter is 15 cm, so length + width = 15.
- Width = 15 − 9 = 6 cm.
- Area = 9 × 6 = 54 cm².
Width = 6 cm; area = 54 cm².
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: Length × width counts square units inside the rectangle. A border follows the outside edges, so it needs perimeter.
Why this fails: A rectangle has two lengths and two widths. You need 2 × length + 2 × width, which is the same as 2 × (length + width).
Why this fails: Perimeter is one-dimensional distance, so use units such as cm or m. Area counts square units, so use cm² or m².
Why this fails: Different side lengths can give the same distance around but different amounts of space inside.
Foundation
Foundation practice
Decide what is being measured, then use the correct operation and unit.
1. A rectangle is 7 cm long and 4 cm wide. What is its perimeter?
2. The same 7 cm by 4 cm rectangle has what area?
3. A square has side length 5 m. What is its perimeter?
4. The same square has what area?
5. A 12 m by 8 m garden needs a fence around the outside. Do you need perimeter or area?
- Perimeter
- Area
6. A classroom floor will be covered with carpet. Do you need perimeter or area?
- Perimeter
- Area
7. Find the perimeter of a 10 cm by 3 cm rectangle.
8. Find the area of a 10 cm by 3 cm rectangle.
9. A rectangle's perimeter should be written in square centimetres (cm²).
- True
- False
10. Area tells how many square units cover a flat region.
- True
- False
Practice
Practice practice
Use the formulas, check your units and notice when the question changes.
11. A rectangle is 14 m long and 6 m wide. Find both perimeter and area.
12. A square has side length 9 cm. Find both perimeter and area.
13. A rectangle has perimeter 34 cm and length 10 cm. What is its width?
14. A 8 m by 5 m patio is being covered with paving stones. Which number matters first: its 26 m perimeter or its 40 m² area?
- 26 m perimeter
- 40 m² area
15. A rectangle is 6 cm by 4 cm. Which pair is correct?
- Perimeter 10 cm; area 24 cm²
- Perimeter 20 cm; area 24 cm²
- Perimeter 24 cm; area 20 cm²
- Perimeter 20 cm²; area 24 cm
16. Maya says a 8 cm by 3 cm rectangle has perimeter 24 cm because 8 × 3 = 24. What should she change?
17. A square has perimeter 28 cm. What is its area?
18. A rectangle has area 48 cm² and length 8 cm. Its width is 6 cm. What is its perimeter?
19. A 15 m by 10 m rectangular garden needs fencing, except for a 2 m-wide gate where no fence is needed. How much fencing is required?
20. A 7 m by 4 m floor is covered with 1 m² tiles. Ignoring cuts and waste, how many tiles are needed?
21. Rectangle A is 10 cm by 2 cm. Rectangle B is 6 cm by 4 cm. Which has the larger perimeter, and which has the larger area?
22. A rectangle has whole-number side lengths and perimeter 20 cm. Give two different possible pairs of side lengths.
Think
Think practice
Compare shapes, reverse the formulas and test claims instead of following a surface pattern.
23. Two rectangles both have area 24 cm². One is 6 cm by 4 cm; the other is 8 cm by 3 cm. Which has the larger perimeter?
24. Two rectangles both have perimeter 24 cm. One is 8 cm by 4 cm; the other is 6 cm by 6 cm. Which has the larger area?
25. If both the length and width of a rectangle are doubled, the area doubles.
- True
- False
26. Liam says a 12 cm by 2 cm rectangle and a 7 cm by 7 cm square have the same perimeter. Is he correct?
27. A rectangle has perimeter 40 cm and length 13 cm. Find its width and area.
28. A rectangle has area 54 cm² and length 9 cm. Find its width and perimeter.
29. A garden is 10 m by 5 m. Its width increases by 2 m while its length stays 10 m. By how much do the perimeter and area increase?
30. Three rectangles each use exactly 32 m of fencing: 10 m × 6 m, 9 m × 7 m, and 12 m × 4 m. Which encloses the greatest area?
Challenge
Challenge practice
Use constraints, search systematically and justify why your answer works.
31. A rectangle has whole-number side lengths and perimeter 24 cm. Which possible rectangle has the greatest area?
32. A 4 cm by 11 cm rectangle has perimeter 30 cm and area 44 cm². Give a different whole-number rectangle with the same perimeter but a larger area.
33. Can two rectangles have the same area but different perimeters? Give an example.
34. Ava says, “If a shape's perimeter doubles, its area must double.” Ben says, “That is not always true.” Who is correct?
35. A rectangular dog run must have area 36 m² and whole-number side lengths. Which dimensions use the least fencing?
36. A rectangle has perimeter 26 cm and whole-number side lengths. Can its area be exactly 42 cm²?
Answer key
- 1. 22 cm
7 + 4 + 7 + 4 = 22 cm. - 2. 28 cm²
7 × 4 = 28 square centimetres. - 3. 20 m
A square has four equal sides, so 4 × 5 = 20 m. - 4. 25 m²
5 × 5 = 25 square metres. - 5. Perimeter
A fence follows the boundary, so the distance around is needed. - 6. Area
Carpet covers the surface inside the floor. - 7. 26 cm
2 × (10 + 3) = 26 cm. - 8. 30 cm²
10 × 3 = 30 cm². - 9. False
Perimeter is a distance, so use centimetres (cm), not square centimetres. - 10. True
Area measures the space inside a 2D shape in square units. - 11. Perimeter = 40 m; area = 84 m²
2 × (14 + 6) = 40, and 14 × 6 = 84. - 12. Perimeter = 36 cm; area = 81 cm²
4 × 9 = 36, and 9 × 9 = 81. - 13. 7 cm
Half the perimeter is 17, so 10 + width = 17. The width is 7 cm. - 14. 40 m² area
The stones cover the surface, so area is the relevant measure. - 15. Perimeter 20 cm; area 24 cm²
2 × (6 + 4) = 20 cm and 6 × 4 = 24 cm². - 16. Use addition around the four sides; the perimeter is 22 cm.
8 × 3 = 24 cm² is the area. Perimeter is 8 + 3 + 8 + 3 = 22 cm. - 17. 49 cm²
Each side is 28 ÷ 4 = 7 cm. Area = 7 × 7 = 49 cm². - 18. 28 cm
2 × (8 + 6) = 28 cm. - 19. 48 m
The full perimeter is 2 × (15 + 10) = 50 m. Subtract the 2 m gate: 48 m. - 20. 28 tiles
The floor area is 7 × 4 = 28 m², so 28 one-square-metre tiles cover it. - 21. A has the larger perimeter; B has the larger area.
A: perimeter 24 cm, area 20 cm². B: perimeter 20 cm, area 24 cm². - 22. Any two different pairs from 1 × 9, 2 × 8, 3 × 7, 4 × 6, or 5 × 5.
Perimeter 20 means length + width = 10. Any whole-number pair that sums to 10 works. - 23. The 8 cm by 3 cm rectangle.
6 × 4 has perimeter 20 cm. 8 × 3 has perimeter 22 cm. - 24. The 6 cm by 6 cm square.
8 × 4 = 32 cm², while 6 × 6 = 36 cm². - 25. False
Doubling both dimensions multiplies area by 2 × 2 = 4. The perimeter doubles, but the area becomes four times as large. - 26. Yes.
The rectangle perimeter is 2 × (12 + 2) = 28 cm. The square perimeter is 4 × 7 = 28 cm. Their areas are different. - 27. Width = 7 cm; area = 91 cm²
Half the perimeter is 20, so width = 20 − 13 = 7. Then 13 × 7 = 91. - 28. Width = 6 cm; perimeter = 30 cm
54 ÷ 9 = 6. Then 2 × (9 + 6) = 30. - 29. Perimeter increases by 4 m; area increases by 20 m².
Old: P = 30 m, A = 50 m². New 10 × 7: P = 34 m, A = 70 m². - 30. 9 m × 7 m, with area 63 m².
Their areas are 60 m², 63 m² and 48 m². Equal perimeter does not force equal area. - 31. 6 cm × 6 cm, with area 36 cm².
Length + width must equal 12. The distinct pairs are 1×11, 2×10, 3×9, 4×8, 5×7 and 6×6. Their areas increase up to 36 cm². - 32. Examples include 5 cm × 10 cm, 6 cm × 9 cm, or 7 cm × 8 cm.
For perimeter 30, length + width = 15. Each listed pair sums to 15 and has area greater than 44 cm². - 33. Yes. For example, 1 cm × 12 cm and 3 cm × 4 cm both have area 12 cm².
Their perimeters are 26 cm and 14 cm, so equal area does not require equal perimeter. - 34. Ben is correct.
A 3 cm square has perimeter 12 cm and area 9 cm². A 6 cm square has double the perimeter, 24 cm, but four times the area, 36 cm². - 35. 6 m × 6 m, requiring 24 m of fencing.
Factor pairs for 36 are 1×36, 2×18, 3×12, 4×9 and 6×6. Their perimeters are 74, 40, 30, 26 and 24 m. - 36. Yes. Its dimensions can be 6 cm by 7 cm.
Perimeter 26 means length + width = 13. Since 6 + 7 = 13 and 6 × 7 = 42, both conditions are satisfied.