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Primary 6 · Circle · Question type 6.8

P6 Circle: Shapes inside shapes

Trace the equal lengths through the nested shapes, then halve the final diameter to find the radius.

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Primary 6, on the MOE Primary Mathematics syllabus.

In the Circle chapter9 question types · see the chapter

The lesson · 8 minutes 39 seconds

How do you solve shapes inside shapes questions in P6?

Watch: the shaded area left when eight quarter circles are taken out of a rectangle

Another worked example

A worked example

The figure shows a straight line AB. The points G, O and H lie on AB so that AG = GO = OH = HB = 12 cm. A semicircle is drawn on AB as diameter. Four smaller semicircles are drawn on AG, GO, OH and HB as diameters, all on the same side of AB as the big one. The part inside the big semicircle but outside the four small ones is shaded. Find the area of the shaded part. (Take π = 3.14.)

The working · Seven lines

The step-by-step working

Written out in full, the way it should appear on paper.

  1. area = π × r × r
  2. AB = 4 × 12 = 48
  3. big radius = 48 ÷ 2 = 24
  4. big semicircle = 1/2 × 3.14 × 24 × 24 = 904.32
  5. small radius = 12 ÷ 2 = 6
  6. four small semicircles = 4 × 1/2 × 3.14 × 6 × 6 = 226.08
  7. shaded = 904.32 - 226.08 = 678.24

The answer is 678.24 cm².

Practise · Modelled on top school exam questions

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Find the shaded parts of two squares

Subtract the two quarter circles from the two squares.

The figure shows a semicircle with AB as diameter and centre O, where AB = 56 cm. The points G and H lie on AB so that AG = GO = OH = HB. A square is drawn on GO and another on OH, both standing inside the semicircle. In each square a quarter circle is drawn with its centre at O and a radius equal to the side of the square. The parts of the two squares outside the quarter circles are shaded. Find the total area of the shaded parts. (Take π = 22/7.)

cm²

Show the step-by-step solution
  1. area = π × r × r
  2. AG = 56 ÷ 4 = 14
  3. two squares = 2 × 14 × 14 = 392
  4. two quarter circles = 2 × 1/4 × 22/7 × 14 × 14 = 308
  5. shaded = 392 - 308 = 84

Ans: 84 cm²

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The last type compares two regions with the same overlap. Difference between Overlapping Regions →

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