A medium question on circles for P5–P6, with a step-by-step solution, diagrams and the answer. A P6 topic. Shaded shapes made of circles and squares are common in the Hua Xia Cup's P6 qualifying round and finals.
The question
In 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?
Show the hints (2)
💡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?
Worked solution
Have a go first, then read the solution step by step.
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.
Play more like this in Numsprite (unlimited questions, instant marking) →A free maths game: every level free from Easy to Expert (★ to ★★★★), unlimited questions, no download