Arc length as the central angle's share of the circumference

What to assume they know

• Why C/D is the same number for every circle — circumference as πd, and therefore as 2πr
• Reflection and rotation as motions that move a figure without changing lengths
• Angles at a point summing to 360°
• Reading a fraction of a turn as a fraction of 360°
• Substituting into a formula and keeping units straight
• The word subtends, for the angle an arc makes at the centre

What they should be able to do

• Convert between the two forms of the circumference formula and say why both are in use
• Prove that a semicircular arc has length πr, using reflection in the diameter, and again using a half-turn about the centre
• Prove that a quarter arc has length πr/2 using a quarter-turn
• Rewrite the half and the quarter as 2πr × 180/360 and 2πr × 90/360 and say what the rewriting is for
• State the general arc-length formula for a central angle θ°
• Justify the general formula from rotation-invariance and additivity, rather than by extending a pattern
• Compute arc lengths from a radius and an angle, and the perimeter of a sector, which is an arc plus two radii
• Compute one lap of a 400 m track from its printed dimensions and account for the 400
• Compute the stagger between adjacent lanes and show it is the same between every adjacent pair
• Decide whether a shorter track needs a different stagger, and justify the answer

Where it usually goes wrong

• "The arc formula is a rule to memorise." It is a fraction of a circumference, and the fraction is written on the page as θ/360. A student who sees the formula as a share will never invert it by accident.
• "The general formula was guessed from two cases." The chapter's own word is guess (p. 126), and two cases would indeed be thin evidence. The formula is forced: equal angles cut equal arcs because rotating the circle about its centre changes nothing, and arcs joined end to end add. Give the argument; the explanation gains its whole spine from it.
• "A sector's perimeter is its arc." It is the arc plus the two radii. The chapter spells this out inside the question (Q4, p. 129) because the mistake is so common.
• "The outer lane runner is disadvantaged by the stagger." She starts further along precisely so that her longer bend is cancelled. The whole point of the computation is that after the stagger everyone runs the same distance.
• "A smaller track needs a smaller stagger." The per-lap stagger is 2π times the lane width and contains no reference to the track's size. This is the single most valuable moment in the topic and it answers the question the chapter opened with on p. 118.
• "The straights matter to the stagger." Every runner runs the same straight distance; the whole difference is on the bends. Setting the straights aside is what makes the arithmetic short.
• "π is 22/7 throughout." The exercise sets instruct 22/7, but the p. 127 track computation uses 3.1416. Mixing them will make the lap fail to come out at 400.

Questions to check understanding

• Arc length from radius and central angle, with π given
• Central angle from arc length and radius
• Perimeter of a sector, with the two radii required
• Length of a semicircular or quarter-circular arc quoted as a fraction of the circumference
• Compute a lap distance for a composite track of straights and semicircular ends
• Compute the stagger for a given lane width, and state whether it changes from lane to lane
• Reasoning answer: explain why the stagger does not depend on the radius of the bend

Examples worth working on the board

Inputs, not answers. Values marked Verified are worked out here; the chapter prints no answers and this volume has no appended answer key.

• Fig. 6.8 (p. 125). A circle with diameter AB drawn horizontally through centre O, radius marked r; the upper semicircular arc is red, the lower is blue. The two colours exist so that a reflection in AB can visibly swap them.
• Fig. 6.9 (p. 125). The same circle with two perpendicular diameters, labelled B at the top, A at the right, C at the left and D at the bottom, the four quarter arcs drawn in four colours. A quarter-turn carries each onto the next.
• The two derivations as printed (p. 125). Reflection in AB exchanges the two semicircular arcs, so they are equal and each is 2πr ÷ 2 = πr. A half-turn about O does the same job. For the quarter, a 90° rotation carries each quarter arc onto another, so all four are equal and each is 2πr ÷ 4 = πr/2.
• The two rewritings (pp. 125–126). πr = 2πr × 180°/360° and πr/2 = 2πr × 90°/360°.
• Fig. 6.10 (p. 126). Centre O with two radii to A and B, the angle at O marked θ°, and the arc AB drawn in red. The general formula stated beside it: arc = 2πr × θ°/360°.
• Exercise Set 6.1 Q3 (p. 129). Two arcs to compute, with π as 22/7: radius 3.5 cm and central angle 60°; radius 6.3 m and central angle 120°. Verified: (60/360)(2 × 22/7 × 3.5) = 22/6 = 3.667 cm; and (120/360)(2 × 22/7 × 6.3) = 39.6/3 = 13.2 m. The second comes out clean, the first does not — say so.
• Exercise Set 6.1 Q4 (p. 129). The perimeter of a sector of a circle of radius 14 cm with sector angle 75°, where the question states explicitly that both straight edges count as well as the curve. Verified: arc = (75/360)(2 × 22/7 × 14) = 55/3 = 18.333 cm; two radii add 28 cm; perimeter 46.333 cm.
• Fig. 6.11 (p. 126) and its dimensions (p. 127). The figure is a coloured overhead view of an eight-lane oval with the number 400 m across the infield and two dots labelled A and B. It carries no dimensions at all — the measurements are in the text on the following page. They are: two straight sections of 84.39 m each; two curved ends that are semicircles about the common centres A and B; innermost semicircle radius 36.5 m; lane width 1.22 m.
• One lap, as the chapter works it (p. 127). The runner is taken to be 0.3 m out from the inner border, so her curve radius is 36.8 m. Two straights give 168.78 m. The two semicircles together are one full circle, of circumference 2 × 3.1416 × 36.8 = 231.22 m. Total 400.00 m. Note that this bullet is the one place in the chapter where π is used as 3.1416 rather than 22/7.
• The stagger, which the chapter asks for and does not compute. Think and Reflect (p. 127) asks for the radius difference between lanes one and two, for the stagger the second-lane runner needs, and whether the stagger between lanes two and three is the same. Verified: the radius difference is the lane width, 1.22 m; the second-lane runner's curve radius is 36.5 + 1.22 + 0.3 = 38.02 m; her lap is 168.78 + 2 × 3.1416 × 38.02 = 168.78 + 238.89 = 407.67 m; the stagger is the difference, 2 × 3.1416 × 1.22 = 7.67 m. The straights contribute nothing to the difference. And yes, the stagger between lanes two and three is the same 7.67 m, because the radius difference is again one lane width.
• The general form of that result, which is the punchline. Verified: 2π(r + w) − 2πr = 2πw. The extra distance per lap depends on the lane width alone and not on r — so it is the same on a tight bend as on a sweeping one.
• Think and Reflect (p. 118), answered. A 200 m track with the same 1.22 m lanes needs the same 7.67 m stagger per lap, because 2πw does not know how big the track is. What changes on the smaller track is how much curve falls inside a single 100 m leg of the relay, so the stagger applied at a particular changeover need not match the 400 m case even though the per-lap figure does. State both halves; the tempting answer is "smaller track, smaller stagger" and it is wrong for the reason that matters.

Figures to have open

• Fig. 6.8 and Fig. 6.9 (p. 125), redrawn with the arcs in contrasting colours so the reflection and the quarter-turn can be shown moving. The colour is not decoration here; it is the argument.
• Fig. 6.10 (p. 126): centre, two radii, θ° and the arc. The chapter's own.
• A dimensioned schematic of the track. This is the figure the chapter does not have. Fig. 6.11 is a picture with no measurements on it, while the Think and Reflect on p. 127 asks students to use Fig. 6.11 to find the stagger. Two straights of 84.39 m, semicircular ends of inner radius 36.5 m about A and B, lane width 1.22 m, and the 0.3 m running line drawn inside lane one.
• A two-lane close-up of one bend, with the 1.22 m gap and the two arc lengths, to carry section 11.
• A 400 m oval and a 200 m oval side by side with the same lane width, for section 12.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics (NCF-SE 2023), Chapter 6, §6.4 "Length of an Arc of a Circle" (pp. 125–126), including the two bulleted rewritings that straddle pp. 125–126.
• The unnumbered subsection "A Closer Look at a 400 m Athletics Track" (pp. 126–127) with its six-bullet lap computation, and the Think and Reflect that closes it (p. 127).
• Figures 6.8, 6.9 (p. 125), 6.10 and 6.11 (p. 126).
• Exercise Set 6.1 Q3 and Q4 (p. 129).
• The chapter's opening page (p. 118) for the relay photograph, the word stagger and the 200 m track question, treated at length in Perimeter as a walk around the border, and why perimeter-to-side ratios are fixed.
• Chapter Summary (p. 154): the arc-length bullet, which names θ as the central angle.
• The same rotation-and-additivity argument delivers the sector-area formula in §6.10.1 (pp. 146–147), handled in A sector's area is its angle's share of the whole. The two videos should make the parallel explicit.