Why C/D is the same number for every circle

What to assume they know

• Perimeter as a walk around the border, and why perimeter-to-side ratios are fixed — perimeter as a walk around a border, and perimeter-to-side ratios holding fixed across a family of scale copies
• Circumference, diameter and radius as named parts of a circle, and D = 2r
• Ratio, and the fact that a common factor cancels from both terms
• Rounding a measurement, and the idea that a measured value carries an error
• Significant figures, at the level of "correct to 3 significant figures"

What they should be able to do

• State the claim of §6.2 precisely: the quotient C/D takes the same value for every circle, whatever its size
• Explain why any two circles are scale copies of one another, and why no other shape family in the chapter is automatically self-similar in this way
• Explain why scaling by a factor k multiplies a curved boundary's length by k, and say what has to be assumed for that sentence to mean anything
• Distinguish the claim that the constant exists from any claim about what it equals
• Carry out the cotton-reel measurement of the chapter's Home Measurement box and state the value it yields, with the reason for wrapping twenty times
• Say what a measurement can establish about C/D and what it can never establish
• Use the constant in both directions: circumference from radius, radius from circumference, and distance travelled from a wheel's diameter
• Answer a ratio question about two circles without evaluating either circumference

Where it usually goes wrong

• "People measured a lot of circles and the answers agreed, so the ratio is constant." Measurement of finitely many circles can never establish a statement about all of them, and every measurement disagrees with every other in the last digit anyway. The constancy is a consequence of similarity; the measuring is how you find out roughly what the constant is.
• "The ratio is constant because π is constant." This is the argument running backwards. π is defined as the common value; you are entitled to that definition only after you know the value is common.
• "Two circles need not be similar — a big circle looks different." Fix a centre and multiply every distance from it by k: a circle of radius r becomes a circle of radius kr, and every circle of every radius is reachable this way. There is only one circle shape. Triangles and rectangles are not like this, which is why no constant of this kind exists for them.
• "Circumference divided by radius is π." It is 2π. The chapter fixes on the diameter (p. 120) and then writes the formula both ways on p. 125. Deciding once which reference length you are using removes most factor-of-two errors in this chapter.
• "Twenty wraps is just to make the thread long enough to handle." It is to divide the reading error by twenty. Say so, because the same trick — repeat and divide — is the whole of experimental technique at this level.
• "A curve's length is obvious, so 'the boundary scales too' needs no comment." The length of a curve is not defined by laying a ruler along it; it is approached by straight pieces. That is the assumption doing the work in section 5, and it is the same idea the polygon method in Cornering π: from inscribed polygons to Mādhava's exact series turns into a computation.

Questions to check understanding

• Circumference from radius or diameter, and the reverse, with π given as 22/7
• The same, reported correct to a stated number of significant figures
• Distance covered by a wheel in a given number of revolutions, and revolutions needed to cover a given distance
• Ratio questions on two circles where the constant cancels and must be seen to cancel
• Explain, in words, why the quotient of circumference to diameter cannot depend on the circle chosen
• Design-and-report question in the chapter's own style: describe a home procedure for estimating the ratio, state its main source of error, and say how you reduced it

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. 6.5 (p. 120). Three circles drawn side by side at increasing size, each labelled C on the boundary and D on the horizontal diameter. This is the figure the whole topic hangs on and it contains no numbers at all — the labels are shared deliberately, so the eye is asked to compare quotients rather than lengths.
• The claim as printed (p. 120). The chapter states that people realised in ancient times that the quotient does not change when the circle's size changes, and immediately asks what its value is. It gives no argument for the first part. Sections 4–6 supply one.
• Home Measurement box (p. 120). You need a cotton reel that has fine thread wound on it. Measure the reel's diameter D as accurately as you can, then wind the thread round it twenty times, unwind, and measure the total length L. Compute L / (20D). The box tells students the thread must be very thin and asks whether their answer lands between 3 and 4, and then whether it lands between 3.1 and 3.2.
• A worked instance, with invented but realistic numbers. Reel diameter 2.4 cm; twenty wraps measure 151 cm. Verified: 151 / (20 × 2.4) = 151 / 48 = 3.1458…, which sits inside the box's tighter band. Use numbers of this size so that the thickness of the thread is visibly the limiting factor.
• The reason for twenty wraps. A single wrap of about 7.5 cm read to the nearest millimetre carries roughly 1.3 % uncertainty; twenty wraps of about 151 cm read to the same millimetre carry roughly 0.07 %. Verified: the absolute reading error is the same, the quantity it is divided into is twenty times bigger. The chapter asks for twenty wraps and does not say why.
• The box's closing question (p. 120). It asserts that the quotient can be estimated by pure geometry with no measuring at all, and asks students to imagine how. The answer is the inscribed-and-circumscribed polygon method, and it is the subject of Cornering π: from inscribed polygons to Mādhava's exact series. Pose it here; do not answer it.
• Exercise Set 6.1 Q2 (p. 129). Circumferences, correct to 3 significant figures, for radii 7 cm, 10 cm and 12 cm, with π taken as 22/7. Verified: 44.0 cm, 62.9 cm (62.857… before rounding) and 75.4 cm (75.428…).
• Exercise Set 6.1 Q1 (p. 129). Perimeter 44 cm, find the radius. Verified: 7 cm.
• Exercise Set 6.1 Q6 (p. 130). A car tyre of diameter 56 cm. How far does the car go in one revolution of the tyre, and how many revolutions in 10 km? Verified: 176 cm per revolution, and 1 000 000 / 176 = 5681.8… revolutions, so about 5682. Note that the second part does not come out whole.
• End-of-chapter Q7 (p. 149). A bicycle wheel of diameter 60 cm rotated 100 times. Verified: 60 × 22/7 × 100 = 18 857.1… cm, about 188.6 m.
• End-of-chapter Q9 (p. 150). A car wheel of outer radius 28 cm: distance in one complete turn, and turns in a journey of 1 km. Verified: 176 cm per turn, and 100 000 / 176 = 568.18…, so about 568 turns.
• Exercise Set 6.1 Q8 (p. 130). Perimeters in the ratio 5 : 4; find the ratio of radii. Verified: 5 : 4. Worth doing precisely because the constant never has to be evaluated — it cancels, which is the thesis in miniature.

Figures to have open

• A single circle enlarged in place through several scale factors, with C and D tracked as numbers beside it and the quotient pinned unchanged. Not in the book; this is the argument the chapter leaves implicit and it needs a picture.
• Fig. 6.5 (p. 120), three circles on one baseline with C and D marked. The chapter's own figure; redraw as a schematic.
• An inscribed regular polygon on a circle, drawn at two scales, to carry section 5. Standard schematic.
• A cotton reel with thread, thread unwound alongside a rule, and the division L / (20D) written out. Redraw from the Home Measurement box (p. 120); the box itself is text and carries no figure.
• A wheel rolling one full turn with the ground marked off in one circumference. Standard schematic, needed for Q6, Q7 and Q9.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics (NCF-SE 2023), Chapter 6, §6.2 "Perimeter of a Circle — The C/D Ratio" (p. 120), together with the closing paragraph of §6.1 that introduces the word circumference (p. 120).
• The boxed HOME MEASUREMENT panel on p. 120 in full, including its two target bands and its closing question about pure geometry.
• Fig. 6.5 (p. 120). §6.2 itself carries no figure of its own — checked on the printed page.
• Exercise Set 6.1 Q1, Q2 (p. 129), Q6, Q8 (p. 130); end-of-chapter Q7 (p. 149) and Q9 (p. 150).
• Chapter Summary (p. 154), first two bullets: π as the constant quotient for all circles, and C = 2πr.
• The value of the constant is §6.2's long historical passage (pp. 120–123), handled in Cornering π: from inscribed polygons to Mādhava's exact series; its irrationality is §6.3 (pp. 123–124), handled in π is irrational, and what that rules out.