Olympiad topic · P5–P6
Circles
A P6 topic. Shaded shapes made of circles and squares are common in the Hua Xia Cup's P6 qualifying round and finals.
Try one
★★ MediumIn the figure, AB is 150 cm long and C is a point on AB (where exactly is not given).
A big semicircle is drawn on AB, and a small semicircle on each of AC and CB (up or down, as shown), making the shaded shape.
(Take π as 3.14.)
What is the perimeter of the shaded shape, in cm?
- Hint 1: Which arcs make up the edge of the shaded shape? Is the line AB part of its edge?
- Hint 2: A semicircle's arc = 3.14 × diameter ÷ 2. What do all the small semicircles' diameters add up to?
- 1.The big semicircle's arc = 3.14 × 150 ÷ 2. No need to work it out yet.
- 2.Each small semicircle's arc = 3.14 × its diameter ÷ 2, whether it faces up or down. Their diameters add up to exactly AB = 150 cm, so all the small arcs together = 3.14 × 150 ÷ 2, as long as the big arc.
- 3.The shaded shape's perimeter = big arc + small arcs = 3.14 × 150 ÷ 2 × 2 = 3.14 × 150 = 471 cm.
- ⚠️Common mistake: adding AB to the perimeter. All of AB is replaced by the small arcs; it is not on the shaded shape's edge.
Answer: 471 cm
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsA semicircle's perimeter, a ring's area.
- ★★MediumFree3 question types · unlimited questionsThe perimeter of several semicircles, a longer rope.
- ★★★HardFree3 question types · unlimited questionsPetals, the path of a rolling circle's centre.
- ★★★★Expert (competition level)VIP3 question types · unlimited questionsCrescents, a tethered cow's grazing area, the area a circle sweeps.
Circles: what it is and how to solve it
Circumference and area of circles, and shapes made of circles, semicircles and squares.
For example: A semicircular lawn of radius 300 m is fenced all round. How long is the fence? The curve is 3.14 × 300 = 942 m, plus the 600 m diameter: 1542 m.
The key idea: Circumference = π × diameter, area = π × radius × radius (π is 3.14 here). For a shaded part, take the small from the large, or fit the pieces together into a whole circle.
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