What will be the length of a semicircle with the same radius r (see the figure)?
Answer: Length of a semicircle = πr (half of 2πr), which can also be written 2πr × 180°360°.
Step-by-step solution
Idea: Reflecting the circle in the diameter AB swaps the red and blue semicircles, so they are equal. Two equal halves make the full circumference 2πr.
- Reflect the circle in the diameter AB (or turn it through 180° about O). The red and blue semicircles swap places, so they have equal length.1 mark
- Together they make the whole circumference 2πr. So each semicircle = 2πr ÷ 2 = πr = 2πr × 180°360°.1 mark
A semicircle of radius r has length πr (that is, 2πr × 180°/360°).
Check: For r = 7 cm: πr = 22 cm, half of the full 44 cm ✓.
Answer to write in the exam
Two semicircles are equal (reflection in AB).
Semicircle = 12 × 2πr
∴ Length = πr
Common mistakes that cost marks
- Giving πr + 2r. That is the perimeter of a semicircular region (arc plus diameter), not the length of the semicircle itself.
- Writing πr2/2, which is the area of a half-disc.
How this can come in the exam
MCQ (1 mark)
The length of a semicircular arc of radius 14 cm is (use π = 227)
- 22 cm
- 44 cm
- 72 cm
- 88 cm
Show answer
(B) 44 cm
πr = 227 × 14 = 44 cm. (72 cm would add the diameter 28 cm.)
Try one yourself
A semicircular arch has radius 3.5 m. Find the length of the curved arch. (Use π = 227.)
Show answer
πr = 227 × 3.5 = 11 m.
More questions like this
- What will be the length of a quarter circle with the same radius (see the figure)?
- The figure depicts a 400 m athletics track. You can see two straight sections of length 84.39 m each, and two curved portions, which are semicircles with a common centre (points A and B); the innermost semicircle on each side has radius 36.5 m. The width of each lane is 1.22 m. Let an athlete make one complete circuit of the track. What is the total distance she runs?
- What is the difference in radius between the first and second lanes? Use the figure to find the stagger needed by the runner in the second lane. Will an equal stagger be needed between the third and second lanes?
- Two circles of equal radius are located such that each circle passes through the centre of the other circle (see the figure). Given that the radius of each circle is r units, find the perimeter of the shape formed by the two circles in terms of r units. (Ignore the dotted portions that lie within the circles.)
- 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.)