The figure shows nine identical rectangles fitted together to make a large rectangle whose area is 72 cm2. Find the perimeter of each small rectangle.
Answer: Let each small rectangle be L × W. Top row: 4L; bottom row: 5W; so 4L = 5W. 9LW = 72 ⇒ LW = 8 ⇒ W2 = 325, W = 4√105 ≈ 2.53 cm, L = √10 ≈ 3.16 cm. Perimeter = 2(L + W) = 18√105 ≈ 11.38 cm.
Step-by-step solution
Given: 9 identical rectangles; Total area 72 cm2
To find: Perimeter of one small rectangle
To find: Perimeter of one small rectangle
Idea: The top and bottom rows have the same width: 4 long sides = 5 short sides. That fixes the shape; the area fixes the size.
- Let each small rectangle have length L and width W. The top row has 4 rectangles lying flat (width 4L) and the bottom row has 5 standing up (width 5W). Both rows are as wide as the big rectangle: 4L = 5W, so L = 5W4.1 mark
- Total area: 9LW = 72 ⇒ LW = 8 ⇒ 5W4 × W = 8 ⇒ W2 = 325 ⇒ W = 4√105 ≈ 2.53 cm.1 mark
- L = 54 × 4√105 = √10 ≈ 3.16 cm.1 mark
- Perimeter = 2(L + W) = 2(√10 + 4√105) = 18√105 ≈ 11.38 cm.1 mark
Each small rectangle is √10 cm × (4√10/5) cm, so its perimeter is 18√10/5 cm ≈ 11.38 cm.
Check: L × W = √10 × 4√105 = 405 = 8 ✓; 9 × 8 = 72 ✓; 4√10 ≈ 12.65 = 5 × 2.53 ✓.
Answer to write in the exam
4L = 5W (widths of the two rows)
9LW = 72 ⇒ LW = 8
54W2 = 8 ⇒ W = 4√105 cm, L = √10 cm
Perimeter = 2(L + W) = 2 × 9√105
∴ Perimeter = 18√105 ≈ 11.38 cm
Common mistakes that cost marks
- Assuming the small rectangles have whole-number sides; here they do not.
- Using 4W = 5L (mixing up which side lies along the row).
- Dividing 72 by 9 and stopping (8 is the area of one rectangle, not its perimeter).
How this can come in the exam
Short answer (3 marks)
Seven identical rectangles, 3 lying flat on top of 4 standing upright, make a large rectangle of area 84 cm2. Find the perimeter of each small rectangle.
Show answer
3L = 4W and 7LW = 84 ⇒ LW = 12 (1 mark); 43W2 = 12 ⇒ W = 3 cm, L = 4 cm (1 mark); perimeter = 2(4 + 3) = 14 cm (1 mark).Try one yourself
Six identical rectangles: 2 lying flat on top of 3 standing up form a large rectangle of area 54 cm2. Find the sides of each small rectangle.
Show answer
2L = 3W, 6LW = 54 ⇒ LW = 9 ⇒ 32W2 = 9 ⇒ W = √6 ≈ 2.45 cm, L = 3√62 ≈ 3.67 cm.
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