Length of an arc · 3 marks

In the figure, we see points P and Q and two paths connecting them. The first path is made up of the semicircle a. The other path is made up of three semicircles (b, c and d). Which path is longer? Choose one: (i) Path a is longer. (ii) Path b + c + d is longer. (iii) The two paths have equal length. (Try to answer this before reading on.)

Answer: (iii) The two paths have equal length. With radii a′, b′, c′, d′: path lengths are πa′ and π(b′ + c′ + d′), and since PQ = 2a′ = 2b′ + 2c′ + 2d′, a′ = b′ + c′ + d′.

Step-by-step solution

Idea: A semicircle’s length is π × radius, so it is proportional to its diameter. The three small diameters add up to the big diameter PQ.

Let the radii of the semicircles a, b, c, d be a′, b′, c′, d′. Length of semicircle a = 12(2π)a′ = πa′.½ mark
Similarly the others have lengths πb′, πc′, πd′, so the second path = π(b′ + c′ + d′).½ mark
The diameters lie along PQ: PQ = 2a′, and also PQ = 2b′ + 2c′ + 2d′. So a′ = b′ + c′ + d′.1 mark
Hence πa′ = π(b′ + c′ + d′): (iii) the two paths have equal length.1 mark

(iii) The two paths have equal length.

Check: Take PQ = 16: a′ = 8, and for example b′ = 4, c′ = 2, d′ = 2. Lengths 8π and (4 + 2 + 2)π = 8π ✓.

Answer to write in the exam

Path 1 = 12 × 2πa′ = πa′

Path 2 = πb′ + πc′ + πd′ = π(b′ + c′ + d′)

PQ = 2a′ = 2b′ + 2c′ + 2d′ ⇒ a′ = b′ + c′ + d′

∴ Path 1 = Path 2; option (iii)

Common mistakes that cost marks

Choosing (ii) because the wiggly path looks longer. The lengths depend only on the total of the diameters.
Thinking it matters that some small semicircles go above and some below PQ. A semicircle’s length does not depend on which side it is drawn.
Using πr² (area) instead of πr (length).

How this can come in the exam

MCQ (1 mark)

A line segment of length 20 cm is split into pieces of 6 cm, 8 cm and 6 cm, and a semicircle is drawn on each piece. The total length of the three semicircles is (use π = 3.14)

15.7 cm
31.4 cm
62.8 cm
20 cm

Show answer

(B) 31.4 cm
Total = π2 × (6 + 8 + 6) = π2 × 20 = 10π = 31.4 cm, the same as one semicircle on 20 cm.

Try one yourself

A semicircle stands on a diameter of 14 cm. Another path consists of seven semicircles, each on a 2 cm piece of the same diameter. Compare the lengths. (Use π = 227.)