Cornering π: from inscribed polygons to Mādhava's exact series

What to assume they know

• Why C/D is the same number for every circle — that circumference divided by diameter is one number for every circle, so there is something to compute
• The Baudhāyana–Pythagoras theorem, used to get the side of a hexagon from a perpendicular
• That a regular hexagon inscribed in a circle has side equal to the radius
• Comparing fractions and decimals, and knowing which of two approximations is nearer a target
• Reading an inequality of the form p < π < q as a trapped value

What they should be able to do

• Explain why a polygon drawn inside a circle gives a lower bound for π, and one drawn outside gives an upper bound
• Show, from the inscribed regular hexagon, that π is greater than 3
• Show, from the circumscribed regular hexagon, that π is less than 2√3, using the Baudhāyana–Pythagoras theorem to find the hexagon's side
• State Archimedes' bracket for π and say what raising the side count achieves and what it cannot achieve
• Place the chapter's named values in time and order them by accuracy, and notice that accuracy did not improve monotonically
• State Mādhava's series for π and explain in what sense it is exact where every earlier value was not
• Compute the first few partial sums of Mādhava's series and describe how they behave
• Explain why the series' exactness does not make it a practical way of getting many digits, and what that implies about how Mādhava reached eleven decimal places
• Account for the symbol π: who introduced it, when, and from which word

Where it usually goes wrong

• "They kept getting better values, so mathematics marched forward." It did not, not in a straight line. Brahmagupta's √10 in 628 CE is a worse approximation than Mesopotamia's 3.125 from about 1900 BCE, and he adopted it knowing what it was for — ease of algebraic manipulation. Accuracy is one goal among several.
• "22/7 is Zu Chongzhi's discovery." The chapter gives 22/7 as his Yuelü (p. 122) and also gives 3 + 1/7, which is the same number, as Archimedes' upper bound seven centuries earlier (p. 121). Both statements are on facing pages. What was new in China was the systematic method that produced 355/113 alongside it.
• "A 96-sided polygon is basically a circle." It is a polygon. Its perimeter is strictly less than the circle's, always, however many sides you take. The method never terminates and never produces an equality — which is precisely why Mādhava's move mattered.
• "An infinite series is just a long approximation." The series is an exact statement: the number π/4 is the limit of those partial sums. Any finite piece of it is an approximation; the series itself is not.
• "Exact means fast." This series is exact and hopelessly slow. Exactness and efficiency are different virtues, and the later names in the chapter's list — Machin, Ramanujan, the Chudnovskys — are all about the second.
• "The polygon method only gives lower bounds." Inscribed polygons give lower bounds; circumscribed polygons give upper bounds. Archimedes' contribution was using both, and Fig. 6.7 shows both in one picture.
• "π was named after a person." The letter was picked in 1706 by Jones because perimetros, the Greek for perimeter, begins with it (p. 123).

Questions to check understanding

• Show that a regular hexagon inscribed in a circle has side equal to the radius, and deduce a bound on π
• Use the Baudhāyana–Pythagoras theorem to find the side of a regular hexagon circumscribing a circle of given radius, and deduce the other bound
• Given two approximations to π, decide which is nearer and by how much
• Given a bracket p < π < q, state the largest possible error in taking either endpoint as π
• Evaluate the first few partial sums of a given alternating series and describe the pattern
• Order a list of historical values of π by accuracy and comment on what the ordering shows about the order of their dates
• Short reasoning answer: explain why no polygon computation can ever give π exactly

Examples worth working on the board

Inputs, not answers. Values marked Verified are worked out here; the chapter prints no answers and this volume carries no appended answer key. Everything attributed to a named mathematician is printed in §6.2, pp. 120–123.

• Fig. 6.6 (p. 121). A regular hexagon inscribed in a circle, with the radius from the centre marked r = 1. The caption asks the reader to see why this forces π > 3.
• The inscribed hexagon bound, worked. A regular hexagon inscribed in a circle of radius r has side exactly r, because each of the six triangles from the centre has two radii and a 60° angle between them and is therefore equilateral. Verified: perimeter 6r = 3D, and the circle's boundary is longer than any polygon inscribed in it, so π = C/D > 3.
• Fig. 6.7 (p. 121). The inscribed hexagon in red, a circumscribed hexagon in blue outside it, the circle between them, r = 1 drawn perpendicular to a side of the outer hexagon with a right-angle mark, and the letter a used three times — on a side of the outer hexagon, and on two segments from the centre to outer vertices. The printed hint is to use the Baudhāyana–Pythagoras theorem.
• The circumscribed hexagon bound, worked. For a regular hexagon the side equals the distance from centre to vertex, both marked a in the figure; the perpendicular from the centre to a side has length 1 and bisects that side. Verified: (a/2)² + 1² = a², so a² = 4/3 and a = 2/√3; perimeter = 6a = 4√3 ≈ 6.928, which over a diameter of 2 gives π < 2√3 ≈ 3.464. The chapter states this bound and leaves the computation to the reader.
• Mesopotamia, about 1900 BCE (p. 121). π taken as 3 + 1/8. Verified: 3.125, low by 0.0166.
• Archimedes, 250 BCE (p. 121). From hexagons, 3 < π < 2√3; from 96-sided polygons, 3 + 10/71 < π < 3 + 1/7. Verified: 3.14085 < π < 3.14286, an interval of width about 0.0020.
• Ptolemy, about 150 CE (p. 122). 377/120. Verified: 3.141666…, high by 0.000074.
• Liu Hui, 263 CE, and Zu Chongzhi, 480 CE (p. 122). The side count was driven as far as 24 576. Zu's two ratios: 22/7 ≈ 3.1428, and 355/113 ≈ 3.1415929. The chapter states that 355/113 was the world's most accurate value for over 800 years, and that no fraction with denominator under 15 000 comes closer. Verified: 355/113 = 3.14159292…, high by about 2.7 × 10⁻⁷; 22/7 is high by about 1.3 × 10⁻³.
• Āryabhaṭa, 499 CE (p. 122). 62832/20000. Verified: exactly 3.1416, high by 0.0000073. He labelled it asanna — approaching, not equal.
• Brahmagupta, 628 CE (p. 122). √10, chosen for algebraic convenience. Verified: 3.16228, high by 0.0207 — which is worse than the Mesopotamian 3.125 of two and a half thousand years earlier, and worse than the 256/81 of the Egyptian area rule the chapter gives on p. 145. It also sits above Archimedes' upper bound of 3 + 1/7 = 3.142857, so it was already known to be out of range when it was adopted. The chapter calls it slightly less accurate and does not say either of these things. This is the strongest single beat in the topic.
• Mādhava of Sangamagrāma (p. 122). π/4 = 1 − 1/3 + 1/5 − 1/7 + …, printed as the first exact formula for π and as the origin of what became calculus. The Chapter Summary (p. 154) prints the same series in the form π = 4(1 − 1/3 + 1/5 − 1/7 + …).
• Partial sums, to show. Verified: one term 4; two terms 2.6667; three terms 3.4667; four terms 2.8952; five terms 3.3397. They straddle π and close in extremely slowly — the gap after n terms is roughly 1/n.
• The consequence, which the chapter does not draw. The chapter says the series let Mādhava compute π to eleven decimal places, and prints 3.14159265358 (p. 123). Verified: eleven correct decimal places from this series alone would need on the order of 10¹¹ terms. So the plain alternating series cannot be what he summed; correction terms and faster related series are what make eleven places reachable.
• After Mādhava (p. 123). Nīlakaṇṭha about 1500, Machin 1706, Ramanujan 1914, the Chudnovsky brothers 1988 extending Ramanujan's ideas; π now known to hundreds of trillions of digits, which the chapter calls neels of digits.
• The boxed note on the symbol (p. 123). William Jones, Welsh, used the Greek letter π for this quotient in 1706, taking it from perimetros; Leonhard Euler made the notation standard.

Figures to have open

• Fig. 6.6 (p. 121): regular hexagon inscribed in a circle, radius marked. The chapter's own; redraw, adding the six radii so the equilateral triangles are visible.
• Fig. 6.7 (p. 121): inscribed and circumscribed hexagons with the same circle, the perpendicular of length 1 and the side a marked. The chapter's own and indispensable — the whole double bound is in this one picture. Redraw with the three a labels distinguished, because in the printed figure they name a side and two centre-to-vertex segments and are easy to confuse.
• An accuracy scale, not a timeline, on which every named value is placed by distance from π. This is an added figure and it is what makes section 9 land.
• A partial-sum plot for Mādhava's series, oscillating either side of π. Not in the book; the chapter prints the series but no picture of it.
• No portraits. The names carry the story; faces do not.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics (NCF-SE 2023), Chapter 6, §6.2's long historical passage under the printed subheading "C/D's Adventurous Journey: From Ancient Approximations to the Exact Formula of Mādhava" (pp. 120–123).
• Figures 6.6 and 6.7, both on p. 121, with their captions and the printed hint.
• The unnumbered green box on p. 123 giving the origin of the symbol π.
• Chapter Summary (p. 154), the bullet listing Archimedes' bracket, Zu Chongzhi's 355/113, Āryabhaṭa's 3.1416 and Mādhava's series.
• Cross-reference inside the chapter: 256/81 as an ancient value bound up with the area of a circle appears on pp. 144–145, and is handled in Slicing a disc into sectors to see where πr² comes from. It is used in section 9 of this brief only as a point of comparison.
• Irrationality, and why there is no best fraction, is §6.3 (pp. 123–124), handled in π is irrational, and what that rules out.