☰ Year plan · Mathematics

Same area, different perimeter, and tens

Week 12 · Same area, different perimeter, and tens · 3 lessons
1

Two rectangles, one area

Discover Mathematics P3: Term 1, Lessons 41–50
GoalShow that rectangles with the same area can have different perimeters, and the other way round.
New words
area · the number of square units inside a shapeperimeter · the distance all the way round a shapelength and width · the two side measures of a rectangle
Explanation

Can two rectangles cover the same space but have fences of different lengths? Let us test with an area of 6 square cm. A rectangle 6 cm by 1 cm has an area of 6 × 1 = 6 square cm and a perimeter of 6 + 1 + 6 + 1 = 14 cm. A rectangle 3 cm by 2 cm has an area of 3 × 2 = 6 square cm and a perimeter of 3 + 2 + 3 + 2 = 10 cm. The areas are equal but the perimeters are not, so the answer is no. The long thin rectangle has the longer perimeter. The opposite is also true: rectangles with the same perimeter can have different areas.

6 cm1 cm3 cm2 cm
Worked examples
RectangleAreaPerimeter
12 by 11226 cm
6 by 21216 cm
4 by 31214 cm
Area 12 square cm. The rectangles 12 by 1, 6 by 2 and 4 by 3 all have an area of 12. Their perimeters are 26 cm, 16 cm and 14 cm. The rectangle that is closest to a square has the smallest perimeter.
RectanglePerimeterArea
9 by 120 cm9
8 by 220 cm16
7 by 320 cm21
6 by 420 cm24
5 by 520 cm25
Perimeter 20 cm. The rectangles 9 by 1, 8 by 2, 7 by 3, 6 by 4 and 5 by 5 all have a perimeter of 20 cm, but their areas are 9, 16, 21, 24 and 25 square cm. The square has the biggest area.
6 cm3 cm9 cm2 cm
Both rectangles have an area of 18 square cm: 6 × 3 = 18 and 9 × 2 = 18. The perimeter of the first is 6 + 3 + 6 + 3 = 18 cm. The perimeter of the second is 9 + 2 + 9 + 2 = 22 cm. The first needs less string.
Class activity

Rectangles on squared paper. Find all the different rectangles that cover exactly 24 square units. Write the length, the width and the perimeter of each. Which one has the smallest perimeter? Which one has the biggest?

Practice
1
Give two different rectangles with an area of 8 square cm and find both perimeters.
2
Both rectangles have an area of 10 square cm. Which has the longer perimeter, and by how much?
3
A rectangle has a perimeter of 20 cm and a length of 7 cm. Find its width and its area.
4
Seif says two rectangles with an area of 12 square cm must have the same perimeter. Is he right? Give an example.
5
Both rectangles have a perimeter of 20 cm: 9 cm by 1 cm and 5 cm by 5 cm. Which has the bigger area?