Farah made a pattern on a square piece of felt of side 20 cm. The pattern is made up of 2 identical big quarter circles and 2 identical small quarter circles, all shaded. Each quarter circle has its centre at a corner of the square. The two big quarter circles are at opposite corners, and the two small quarter circles are at the other two corners. Along each side of the square, the big quarter circle and the small quarter circle are 2 cm apart. The radius of each big quarter circle is twice the radius of each small quarter circle. (Take π = 3.14)
- What is the radius of the small quarter circle?
- What is the total perimeter of the shaded part?
Need a hint?
For (a), look at one side of the square. The two radii and the 2 cm gap fill the whole side. For (b), each shaded piece is separate, so find its perimeter and then add them up.
Worked answer
- (a) Along one side: big radius + gap + small radius = 20 cm.
- Big radius + small radius = 20 − 2 = 18 cm.
- Big radius = 2 units and small radius = 1 unit, so 3 units = 18 cm.
- 1 unit = 18 ÷ 3 = 6 cm, so the small radius is 6 cm.
- (b) Big radius = 2 × 6 = 12 cm.
- Perimeter of 1 big quarter circle = 12 + 12 + (1/4 × 2 × 3.14 × 12) = 24 + 18.84 = 42.84 cm.
- Perimeter of 1 small quarter circle = 6 + 6 + (1/4 × 2 × 3.14 × 6) = 12 + 9.42 = 21.42 cm.
- Total perimeter = 2 × 42.84 + 2 × 21.42 = 85.68 + 42.84 = 128.52 cm.
The mistake this question tests
Area and perimeter formulas
Many pupils find only the curved parts and forget that each quarter circle also has two straight edges (its two radii), so the perimeter is too small.
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