Applications only. Radians are not required at Standard Level.
Run it to the 360° setting and stop. The arc reads 37.70. Then ask the class what the 120° sector's area was. It was 37.70 as well. That is the lesson, and it lands in about four seconds because they wrote 37.70 down themselves two minutes earlier.
Then the r = 2 setting, where the arc and the area are both 4.19. Ask whether it is a fluke of 120°. Try 75° on the board: 2.62 and 2.62. The point to name is that 2r and r² are both 4 when r is 2, so at that radius the two formulas agree at every angle.
What the two together establish is the habit: what the question asked for decides the formula, never what the number looks like.
| Question | Answer |
|---|---|
| 1. Arc, r = 9, 140° | (140/360) × 2π × 9 = 7π = 21.99 cm. |
| 2. Area, same sector | (140/360) × π × 81 = 98.96 cm². |
| 3. Perimeter, same sector | 21.99 + 9 + 9 = 39.99 cm. |
| 4. Angle, r = 5, area 30 | 30 = (θ/360) × 25π, so 137.5°. |
| 5. The student who answered 4.19 for the area | B. Right, right at every angle for r = 2, and the working still has to be shown. |
1 markThe correct formula, with the θ/360 present.
1 markCorrect substitution.
1 markThe answer, with units.
Units earn their mark here more often than anywhere else in Topic 3, because the number genuinely does not say which quantity it is. Insist on cm and cm² from the first example, not as pedantry but because it is the only label the answer has.
| They wrote | What happened |
|---|---|
| 56.55 | On question 1, the whole circumference. The 140/360 is missing. At least this one looks too big. |
| 98.96 for the arc | Used πr². They have the fraction and the wrong circle formula. |
| 254.47 | On question 2, the whole circle's area. Same omission as 56.55. |
| 21.99 for the perimeter | The arc on its own. The commonest error in this sub-topic. Ask them to trace the shape with a finger. |
| 30.99 | On question 3, one radius instead of two. |
| 222.5° | On question 4, the reflex sector. Worth a word: the question does not say which, and 30 cm² out of a total 78.54 has to be the smaller one. |
| 343.8° | On question 4, solved the ARC equation. Nearly a whole circle, which is a usable self-check. |
"Why isn't the perimeter just the arc?" Because a perimeter goes all the way round a closed shape and the sector is closed by two straight edges. Draw a flower bed and ask how much edging to buy.
"Can I use ½r²θ?" Not at Standard Level on this course, because that version needs θ in radians and radians are not required here. It is sub-topic 3.7, Higher Level. A student who has met it elsewhere should be told it is the same formula with the fraction already folded in.
"Which sector does the question mean?" Usually the one drawn, and if nothing is drawn, the minor one. If an area or arc is given, check it against the whole circle: more than half means the reflex one.
| Stage | What to do |
|---|---|
| Demonstrate | Type the fraction FIRST, so 140/360*2π*9 rather than 2π*9*140/360. Both give 21.99, but the first makes an omission visible as you type it. Then do the reverse problem with the solver, which is the only part of this sub-topic the machine does anything interesting in. |
| Where they stick | The Casio's Equation Solver wants the variable to exist in the expression before it will solve; students type the equation with the angle already evaluated and get nothing to solve for. On the Nspire, nSolve needs the variable named as the second argument. |
| The check | Before anything is typed, ask roughly what fraction of the circle this is. 140/360 is a bit under half, so the arc should be a bit under half of 56.55. Anyone who then writes 56.55 has ignored their own estimate. |
Ask which machine each student has before the lesson, not during it. This sub-topic is one of the few in Topic 3 where degrees mode does not matter, because no trigonometric function is involved.
| Step | What |
|---|---|
| 1 | The sprinkler, cold. Ask for the watered edge and the watered ground. Collect both. |
| 2 | Run the figure to 360°. Let 37.70 appear twice. |
| 3 | Build the two formulas from "fraction of the whole circle", not as two things to learn. |
| 4 | Perimeter, with a shape traced on the board. |
| 5 | The r = 2 setting, then 75° on the board to show it is not about the angle. |
| 6 | Backwards from an area, by hand and then on the solver. |
Do not say "the area is always bigger". It is not a comparison that means anything, the numbers are in different units, and at r = 2 they are equal. Saying it gives the class a false check they will rely on under pressure.
Do not introduce radians here to save time later. They are not required at Standard Level on this course, and a student carrying ½r²θ into a degrees question will put 140 where 2.44 belongs and be out by a factor of 57.