A sprinkler on a school field in Bangkok throws water 6 m and sweeps through 120°. The watered edge is 12.57 m long and the watered ground is 37.70 m². Forget the 120/360 on the edge and you get 37.70, which is the other answer to the same question.
Both formulas are the whole circle times θ/360. That is the only idea here. Everything that goes wrong goes wrong because the number that comes out looks reasonable either way.
The shaded part is the sector. Its perimeter is the curved edge plus the two straight ones.
For a radius r and an angle θ in degrees:
The third one is not a formula anybody gives you. If the question asks for the perimeter, or for the length of fencing, or for the edging round a flower bed, the two straight sides are part of it. Leaving them out here halves the answer.
The dangerous slip. The full circumference of this circle is 2π × 6 = 37.70. The sector's area is also 37.70. So a student who writes the arc as 2πr and forgets the θ/360 produces a number that is sitting in the correct answer to the other part of the question, and it looks like a win. The only defence is to ask whether the fraction is in there, every time.
People try to tell an arc from an area by which is bigger. At r = 2 that fails completely:
And it is not a coincidence of 120°. At 75° both come to 2.62, and at any angle at all they are equal, because 2r and r² are the same number when r = 2. What tells you which formula to use is what the question asked for. Nothing else.
A sector of radius 5 cm has an area of 30 cm². Find the angle.
Under 180°, so it is the smaller of the two sectors, which is worth saying out loud because a question can mean either one.
Radians are not required at Standard Level on this course. If you have seen ½r²θ somewhere, that is the Higher Level version with θ in radians, and it is sub-topic 3.7. At SL, work in degrees and keep the θ/360.
Forwards you just type the formula. Backwards, the machine will solve the equation for you, and the only part you have to get right is writing it down.
When you may use it. Applications. A calculator is allowed in every paper, so there is no version of this you have to do by hand.
The mark people lose. Typing the whole-circle formula and forgetting the fraction. At r = 9 the arc is 21.99 and the full circumference is 56.55, so that one at least looks wrong. At r = 6 it does not: the full circumference is 37.70 and the sector area is 37.70, and the two halves of the question seem to agree with each other. Put the θ/360 in first, before the rest of the expression, so you cannot leave it out.
1. A sector has radius 9 cm and angle 140°. Find the arc length, to 2 decimal places.
2. For the same sector, find the area, to 2 decimal places.
3. For the same sector, find the perimeter, to 2 decimal places.
4. A sector of radius 5 cm has area 30 cm². Find the angle, to 1 decimal place.
5. A student is told a sector of radius 2 has an arc of 4.19 and is asked for its area. They answer 4.19. What should you say?
1 markThe right formula, with the θ/360 in it.
1 markCorrect substitution.
1 markThe answer, with units: cm for an arc, cm² for a sector.
The units carry a mark on their own here more often than anywhere else in the topic, precisely because the numbers do not tell you which quantity you have found.
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