A robot arm 6 m long rotates through π/3, and its tip wipes out 18.85 m² of floor. Put 60 into the same formula instead and it reports 1080, which is nine and a half times the whole circle the blade could possibly reach.
The formulas have no θ/360 in them. That is the whole reason radians exist, and the whole reason degrees break them.
One radian is the angle at the centre of a circle for which the arc is as long as the radius. That is the definition, and everything else drops out of it.
Convert by multiplying by π/180 or by 180/π. Degrees to radians, multiply by π/180. Radians to degrees, multiply by 180/π. If you are unsure which, check against something you know: 90° must come out near 1.6, not near 5000.
| In degrees | In radians | |
|---|---|---|
| Arc length | (θ/360) × 2πr | rθ |
| Sector area | (θ/360) × πr² | ½r²θ |
The radian versions are what you get when the θ/360 and the 2π cancel. For the arm, r = 6 and θ = π/3:
Both exact, both one multiplication. This is the whole advantage, and it is why the rest of Higher Level assumes radians.
rθ and ½r²θ are only true for θ in radians. Put 60 in and you get:
Arc: 6 × 60 = 360 m. Area: ½ × 36 × 60 = 1080 m².
Both are 57.3 times too big, because that is 180/π. The area one is the easier to catch: the whole circle is only 113.1 m², so 1080 is nine and a half whole circles, from a sector that is a sixth of one.
So the check is nearly free. A sector area has to be smaller than πr², and an arc has to be shorter than 2πr. Those two sentences cost a glance and catch every angle bigger than 6.28°, which is almost every angle a question will give you. Below that it is quiet: 5° put into ½r²θ at r = 6 gives 90 m², comfortably under the circle's 113.1, and nothing warns you. So the check is a safety net, not a substitute for putting the angle in radians in the first place.
The radian formulas rearrange without any fuss, which is the other reason they are worth having.
Higher Level papers assume radians unless a question carries a degree symbol, so the mode is a decision you make per question rather than once per topic.
When you may use it. Applications. A calculator is allowed in every paper. Analysis does not allow one in Paper 1, and radian work there is left in exact multiples of π rather than evaluated.
The mark people lose. Leaving the machine in degrees from Topic 3's Standard Level work. The IB rule is that a Higher Level paper assumes radians unless a degree symbol appears, and plenty of them appear: 3.8 does the whole ambiguous case in degrees, 3.10 resolves forces at 60° and 120°, and 3.13 reports an angle of 27.3°. So the mode is set by the question in front of you, not by where you are in the topic, and nothing reminds you. The symptom is an answer that is 57.3 times too big or too small, so if a sector comes out larger than its circle, check the mode before you check the arithmetic.
1. Convert 60° to radians, to 3 decimal places.
2. A sector has radius 6 and angle π/3. Find its area, to 2 decimal places.
3. For the same sector, find the arc length, to 2 decimal places.
4. One radian is how many degrees? Give it to 1 decimal place.
5. A student computes a sector area of 1080 m² on a radius of 6 m. What is the quickest way to know it is wrong?
1 markThe conversion, if the question mixes units.
1 markThe correct formula, in its radian form.
1 markThe answer, exact or to the stated accuracy.
An exact answer in terms of π is usually accepted and sometimes asked for, so 6π is worth writing down beside 18.85. It also makes a mode error impossible to hide.
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