As the slices become smaller and smaller, the arcs in the figure (B) become more and more closer to a line. This makes the figure more and more closer to a parallelogram with base = half the circumference (why?) = πr, height = radius r.
Answer: The slices alternate: half of them point up and half point down. The arcs of the up-pointing slices make the bottom edge and the arcs of the down-pointing slices make the top edge. So each long edge gets half of all the arcs, i.e. half of 2πr = πr.
Step-by-step solution
Idea: The whole circumference 2πr is shared out equally between the top and bottom edges of the rearranged shape.
- The arcs of all the slices together make the full circumference, 2πr.1 mark
- In the rearrangement the slices point alternately up and down, so exactly half the arcs lie along the bottom and half along the top. Each long side therefore has length 12 × 2πr = πr. The slanting sides are radii, so the height is r, and area = πr × r = πr2.1 mark
Half the slices’ arcs form the base (the other half form the top), so the base is half of 2πr, which is πr.
Answer to write in the exam
Total length of arcs = 2πr
Slices alternate ⇒ half the arcs on the base, half on the top
Base = 12 × 2πr = πr; height = r
∴ Area = πr × r = πr2
Common mistakes that cost marks
- Taking the base as the full circumference 2πr, which would give area 2πr2.
- Taking the height as the diameter.
How this can come in the exam
MCQ (1 mark)
A circle of radius 7 cm is cut into many thin slices and rearranged into a near-parallelogram. Its base is about (use π = 227)
- 7 cm
- 14 cm
- 22 cm
- 44 cm
Show answer
(C) 22 cm
Base = πr = 227 × 7 = 22 cm.
Try one yourself
A circle of radius 14 cm is cut into slices and rearranged as in the figure. Find the base and height of the near-parallelogram and its area. (Use π = 227.)
Show answer
Base πr = 44 cm, height 14 cm, area 616 cm2.
More questions like this
- Find the area of a sector of a circle with radius 7 cm if the angle of the sector is 60°.
- Find the area of a quadrant of a circle whose circumference is 44 cm.
- The length of the minute hand of a clock is 7 cm. Find the area swept by the minute hand in 10 minutes.
- A chord of a circle of radius 10 cm subtends 90° at the centre. Find the area of the corresponding: (Use π ≈ 3.14.)
- A chord of a circle of radius 15 cm subtends an angle of 60° at the centre of the circle. Find the areas of the corresponding minor and major segments of the circle. (Use π ≈ 3.14 and √3 ≈ 1.73.)