Olympiad topic · P4–P6
Perimeter and area
Common in the Hua Xia Cup's P4 to P6 papers; harder ones work out areas from the ratios between triangles' areas.
Try one
★★ Medium5 identical rectangular tiles make a big rectangle, as shown:
1 lying down in the top row and 4 standing up in the bottom row, with no gaps.
The big rectangle has a perimeter of 54 cm.
What is the area of the big rectangle, in cm²?
- Hint 1: The top and bottom rows are the same length. How do a tile's length and width compare?
- Hint 2: Call a tile's length 4 parts and its width 1. How many parts is the big rectangle's perimeter?
- 1.Both rows are the same length: 1 tile length = 4 tile widths, so length : width = 4 : 1.
- 2.Call the length 4 parts and the width 1. The big rectangle is 1 × 4 = 4 parts long and 4 + 1 = 5 parts wide.
- 3.The big rectangle's perimeter is (4 + 5) × 2 = 18 parts, so one part = 54 ÷ 18 = 3 cm.
- 4.The big rectangle is 12 cm by 15 cm: area 12 × 15 = 180 cm².
- ⚠️Common mistake: sharing out the perimeter or area before finding how the tile's length and width compare. The two rows being equal fixes that ratio.
Answer: 180 cm²
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsPerimeter by sliding, shapes put together.
- ★★MediumFree3 question types · unlimited questionsSides from a perimeter, overlaps, a picture frame.
- ★★★HardFree3 question types · unlimited questionsA rectangle cut into pieces, a trapezium's diagonals, areas from ratios.
- ★★★★Expert (competition level)VIP3 question types · unlimited questionsSeveral triangles' area ratios worked from each other.
Perimeter and area: what it is and how to solve it
Perimeters and areas, often by cutting, moving or completing a shape into a familiar one.
For example: A staircase shape, 10 cm along the bottom and 7 cm at its highest, all corners square. Its perimeter? Slide the steps' edges out to make a rectangle: (10 + 7) × 2 = 34 cm.
The key idea: For perimeter, slide the steps and dents to the outline of a rectangle; for area, cut and fill or take the small from the large. Triangles of the same height have areas in the ratio of their bases.
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