What will be the length of a quarter circle with the same radius (see the figure)?
Answer: Length of a quarter circle = πr2 (a quarter of 2πr), also written 2πr × 90°360°.
Step-by-step solution
Idea: Turning the circle through 90° about its centre moves each quarter onto the next one, so all four quarters are equal.
- Rotate the figure through 90° about O. Each quarter circle lands exactly on the next one, so all four have the same length.1 mark
- Four equal quarters make 2πr, so each one = 2πr ÷ 4 = πr2 = 2πr × 90°360°.1 mark
A quarter circle of radius r has length πr/2 (that is, 2πr × 90°/360°).
Check: For r = 14 cm: πr2 = 22 cm, and 4 × 22 = 88 cm = 2π × 14 ✓.
Answer to write in the exam
Four quarter circles are equal (quarter-turn symmetry).
Quarter circle = 14 × 2πr
∴ Length = πr2
Common mistakes that cost marks
- Dividing by 2 instead of 4.
- Adding the two radii. They belong to the perimeter of a quadrant region, not to the arc.
How this can come in the exam
MCQ (1 mark)
The length of a quarter circle of radius 21 cm is (use π = 227)
- 16.5 cm
- 66 cm
- 132 cm
- 33 cm
Show answer
(D) 33 cm
14 × 2 × 227 × 21 = 1324 = 33 cm.
Try one yourself
Find the length of a quarter circle of radius 7 cm. (Use π = 227.)
Show answer
14 × 2 × 227 × 7 = 11 cm.
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