Perimeter does not measure area
Two shapes can have the same edge and different insides. Count the unit squares.
Rangoli powder
Two rectangles are coloured evenly with rangoli powder. A is 7 cm by 4 cm and B is 8 cm by 3 cm. A thread laid round the edge of A is as long as a thread round the edge of B: 22 cm each.
Which rectangle needs more powder?
What this lesson covers
The idea
The perimeter of a region is not indicative of its area: regions can have the same perimeter but different areas, and vice versa, and a region with a larger perimeter can even have a smaller area.
Rangoli powder
Two rectangles are coloured evenly with rangoli powder. A is 7 cm by 4 cm and B is 8 cm by 3 cm. A thread laid round the edge of A is as long as a thread round the edge of B: 22 cm each.
Which rectangle needs more powder?
- A, 7 cm by 4 cm
- B, 8 cm by 3 cm
- Both the same, the threads are equally long
Keep rectangles
Drag the corner knob to make a rectangle, and keep the ones you like. The tray looks for pairs and ticks each discovery. Find all three.
The edge does not tell the inside
The perimeter of a region is not indicative of its area: regions can have the same perimeter but different areas, and vice versa, and a region with a larger perimeter can even have a smaller area.
The powder needs the area: 7 × 4 = 28 unit squares against 8 × 3 = 24. The edges are equally long: 2 × (7 + 4) = 22 and 2 × (8 + 3) = 22. Same perimeter, different areas.
It works the other way too. Area 24 can have perimeter 20 (6 × 4), 22 (8 × 3) or 28 (12 × 2). Same area, different perimeters.
A larger perimeter can even have a smaller area: 12 × 2 has perimeter 28 but area 24, while 7 × 4 has perimeter 22 but area 28. That is why we measure area by counting unit squares.
Notes
The perimeter of a region is not indicative of its area: regions can have the same perimeter but different areas, and vice versa, and a region with a larger perimeter can even have a smaller area.
Check yourself
Both rectangles have the same perimeter, 24 cm. Which one has the bigger area?
A rectangle has an area of 36 cm² and is 9 cm long. What is its perimeter, in cm?
Answer: 26
Breadth = 36 ÷ 9 = 4 cm. Perimeter = 2 × (9 + 4) = 26 cm.
Rectangle P is 12 cm by 2 cm and rectangle Q is 7 cm by 4 cm. Which statement is true?
Ravi says, "The fence round my field is longer than the fence round yours, so my field is bigger." Is he right?
- They have the same area. The same perimeter does not mean the same area. Count the squares: 10 × 2 = 20 and 6 × 6 = 36.
- A, the long thin one. A has 10 × 2 = 20 unit squares. B has 6 × 6 = 36.
- B, the square — correct. Yes! B has 6 × 6 = 36 unit squares and A has only 10 × 2 = 20, although both edges are 24 cm long.
- P has the larger perimeter and the larger area. P has area 12 × 2 = 24, but Q has 7 × 4 = 28. Q has the larger area.
- P has the smaller perimeter and the smaller area. P has perimeter 2 × (12 + 2) = 28, and Q has 2 × (7 + 4) = 22. P has the larger perimeter.
- Q has the larger perimeter but the smaller area. It is the other way round: P has perimeter 28 against 22, and area 24 against 28.
- P has the larger perimeter but the smaller area — correct. Yes! P: perimeter 28, area 24. Q: perimeter 22, area 28. A larger perimeter can have a smaller area.
- Not always: a long thin field can have a longer fence but less area — correct. Yes! A 12 m by 2 m field has a 28 m fence and 24 m² of land. A 7 m by 4 m field has a 22 m fence and 28 m².
- Yes, a longer boundary always means a bigger area. Perimeter does not measure area. A long thin field has a long fence and little land.
- No, a longer boundary always means a smaller area. That is not true either. The perimeter simply does not tell the area. Count the unit squares.