Analysis meets radians at Standard Level, unlike Applications, where they are Higher Level only. From here on this topic is in radians, and Paper 1 wants the exact forms.
Step through all four regions and leave the three-number list on the right up the whole time. 18.85, 15.59, 3.26. The point is not any one of them; it is that three words name three areas and two of them are nearly the same size while the third is nothing like either.
On the segment view, ask what fraction of the sector the triangle took. 83%. That number is why giving the sector for the segment is not a near miss, and it is worth having them work it out rather than hearing it.
The arc view is the one that catches the perimeter error: the arc is drawn thick and the two radii are drawn warm, so the shape has three edges on the screen. Trace it with a finger on the board.
| Question | Answer |
|---|---|
| 1. The sector | ½ × 36 × π/3 = 6π = 18.85. |
| 2. The triangle | ½ × 36 × sin(π/3) = 9√3 = 15.59. |
| 3. The segment | 6π − 9√3 = 3.26. |
| 4. The perimeter | 6.28 + 6 + 6 = 18.28. |
| 5. Why the sector is larger | B. θ = 1.047 against sinθ = 0.866, and the triangle sits inside. |
Questions 1 to 3 are deliberately a chain, so the subtraction is the only new step by the time they reach it. Insist on both areas written down before the subtraction; a student who does it in one go on the calculator loses the structure and usually the mark.
1 markThe correct formula for the region asked for.
1 markThe angle in radians.
1 markThe subtraction, on a segment.
1 markThe answer, exact on Paper 1 and with units.
The subtraction is its own mark, which is worth telling them: finding both areas and forgetting to subtract scores two out of four rather than nothing, so the working is never wasted.
| They wrote | What happened |
|---|---|
| 18.85 for the segment | The error this page exists for. They read the diagram and not the word. Send them to the 83%. |
| 37.70 on question 1 | r²θ, with the half dropped. Note it is also the circumference of this circle, a coincidence of r = 6; say so, or someone will build a theory on it. |
| 15.59 on question 1 | Used sinθ where θ belongs. The two formulas differ by one character and they have picked the wrong one. |
| 1080 | Degrees in the radian formula. The whole circle is 113.1, so it is nine and a half circles, and a size check kills it instantly. |
| 31.18 | r²sinθ on question 2: the half dropped again, and exactly twice the answer. |
| −3.26 | Triangle minus sector. Ask which region is inside which. |
| 34.44 | The two areas added. A segment is a difference. |
| 6.28 on question 4 | The arc alone. The commonest perimeter answer anywhere in the topic. |
| 12.28 | Arc plus one radius. Half way, and a finger on the diagram fixes it. |
"Why radians at all?" Because of two things coming later. The formulas lose their θ/360 fractions, which is cosmetic; and the derivative of sin x is cos x only in radians, which is not. The calculus reason is the honest one and it is worth giving even though Topic 5 is months away, because otherwise radians look like a second system invented to be annoying.
"Do I have to give the exact form?" On Paper 1 yes, and the question will be built so that it exists: 6π, 9√3, 6π − 9√3. On Paper 2 a decimal to three significant figures is fine. The practical advice is to work exactly and convert at the end if asked, because exact working never loses accuracy and decimals accumulate it.
"Is the segment formula ½r²(θ − sinθ)?" Yes, and it is worth deriving in front of them by factorising the subtraction rather than handing it over. A student who has seen it come out of sector minus triangle can rebuild it; a student who has memorised it will use it on a sector question.
"Which angle do I use if the question gives degrees?" Convert first, then use the formula. The common failure is converting halfway through, so insist the conversion is line one of the working. π/3 written down early is also the thing that makes the exact answer available at the end.
| Demonstrate | Type the sector with a DECIMAL half, 0.5*6^2*π/3, and get 18.8496. Then type it with the fraction template and get 6π. Same mathematics, two different answers on the screen, and only one of them earns a Paper 1 mark. That contrast takes thirty seconds and explains the whole exact-form requirement. Use the x² key or the right arrow to leave the exponent: typing ^ opens a superscript box and everything after it stays inside. |
|---|---|
| Where they stick | Believing the mode matters here. It does not for the sector or the arc, because neither formula contains a trigonometric function, and students who have been told "always check radians" are confused when degrees mode gives the right answer. It matters for the TRIANGLE, where the sine is, and that is the one to point at. |
| The check | Against the whole circle. πr² = 113.1 and 2πr = 37.70 here, so any sector area above 113.1 or arc above 37.70 is wrong before it is checked. It catches the dropped half and the degrees error, both. |
A decimal anywhere in the entry forces a decimal answer on both machines, so 0.5 can never produce 6π. Teach the fraction template as the default in this topic.
| Step | What |
|---|---|
| 1 | A radian from the definition: the angle whose arc is one radius. Count 2π of them round the circle. |
| 2 | Both conversions, from 180° = π, not as two fractions to learn. |
| 3 | Arc and sector, derived as fractions of the whole circle, then simplified. |
| 4 | Both on r = 6, θ = π/3, exactly and then as decimals. |
| 5 | Now ask for the area between the chord and the arc. Collect the 18.85. |
| 6 | Figure, the triangle view, then the segment view. The 83%. |
| 7 | Perimeter, traced with a finger. |
| 8 | Backwards from an area, with the under-2π check. |
Do not say "a segment is half a sector". Nobody means to say it, but "take the triangle off and you have about half" slips out, and here the segment is a sixth of the sector. The fraction depends entirely on the angle: at small angles the segment is a tiny sliver, and at π it is the whole semicircle.
Do not say "always put the calculator in radians". It is good advice and it teaches the wrong model, because students then think the sector formula depends on the mode. Say that the mode matters wherever a sine, cosine or tangent appears, which is true, checkable and tells them where to look.