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Most students meet π as "3.14" or "22/7" and never ask how anyone found more digits. In 1424 a Persian astronomer named Jamshid al-Kashi pushed the value further than anyone before him, and he did it with nothing more than circles, polygons and very careful square roots. This post tells his story, shows the method with numbers you can check on a calculator, and ends with what his way of working can teach you.
Who was Jamshid al-Kashi?
According to the MacTutor biography from the University of St Andrews, al-Kashi was born around 1380 in Kashan, in present-day Iran, and grew up in poverty during Timur's wars. He dated many of his works to the day, which is why we know so much about him. His first datable act is an observation of a lunar eclipse in Kashan on 2 June 1406.
His life changed when Ulugh Beg, the ruler of Samarkand and a serious astronomer himself, invited him to the school of learning he was building there, along with around sixty other scientists. Construction of Ulugh Beg's observatory began in 1424. Al-Kashi wrote letters home to his father describing scientific meetings where hard problems were set, and on a couple of occasions only he solved them. He died in Samarkand on 22 June 1429.
What did al-Kashi want to measure?
Al-Kashi was not chasing digits for fun. Astronomers needed precise tables, and a small error in π spreads into every calculation built on it. A paper by Mohammad K. Azarian in Mathematics Interdisciplinary Research (2019) describes how he set his goal. He imagined a circle whose radius was six hundred thousand times the radius of the Earth, and asked for a polygon whose perimeter would differ from that circle's circumference by less than the width of a horse's hair.
Only then did he work out how many sides that needed. The answer was polygons with 3 × 2^28 = 805,306,368 sides, inscribed in and drawn around the circle. The result, completed in Samarkand in July 1424, was his Treatise on the Circumference. MacTutor states that he found 2π to nine sexagesimal (base-60) places and converted it into sixteen decimal places; Azarian counts seventeen with rounding. Either way, MacTutor notes it would be almost 200 years before Ludolph van Ceulen went further, with 20 decimal places.
How does the polygon method work?
The idea goes back to Archimedes. A regular polygon drawn inside a circle has a perimeter a little less than the circumference. Double the number of sides and the gap shrinks. Al-Kashi's contribution, as Azarian explains, was a theorem that let him get each new side length from the previous one quickly and accurately.
Here is the same doubling with a circle of radius 1, so the circumference is 2π and π is half the perimeter. If a regular polygon has side s, the polygon with twice as many sides has side
s′ = √(2 − √(4 − s²))
You can check every line below on a calculator.
Hexagon (6 sides). Each side equals the radius, so s = 1. Perimeter = 6, giving π ≈ 6 ÷ 2 = 3.
12 sides. s′ = √(2 − √3) = √(2 − 1.73205) = √0.26795 ≈ 0.51764. Perimeter ≈ 12 × 0.51764 = 6.2117, so π ≈ 3.1058.
24 sides. s² ≈ 0.26795, so 4 − s² = 3.73205 and its square root is 1.93185. Then s′ = √(2 − 1.93185) = √0.06815 ≈ 0.26105. Perimeter ≈ 24 × 0.26105 = 6.2653, so π ≈ 3.1326.
Keep going. At 96 sides the inscribed polygon gives π ≈ 3.1410. The polygon drawn around the circle gives an upper value, about 3.1427 at 96 sides, so π is trapped between the two.
Starting from the hexagon, 27 doublings give 6 × 2^27 = 3 × 2^28 sides, which is al-Kashi's polygon. The hard part is that every square root has to be carried to many more places than the final answer, or rounding errors pile up. He did this by hand, working in base 60.
What mistakes should you avoid when you try this?
Rounding too early. If you round each square root to three places, by the 48-gon your answer drifts. Keep at least two more digits than you want in the result.
Mixing up radius and diameter. With radius 1 the perimeter approaches 2π, not π. Halve it.
Forgetting the upper bound. One polygon gives a value that is too small. Without the outside polygon you cannot say how close you are.
What else did he leave behind?
His Key to Arithmetic, completed on 2 March 1427, was a large textbook for students in Samarkand, covering what astronomers, surveyors, architects and traders needed. It uses decimal fractions clearly. MacTutor is careful here: historians once called al-Kashi the inventor of decimal fractions, but the idea was already present in earlier work by al-Samawal, and Rashed's view is that al-Kashi's real step was to treat decimal and sexagesimal fractions as parallel systems and to use decimals for numbers like π. Our post on Simon Stevin and decimals picks up the European side of that story. For another route to π, read about Zu Chongzhi and the fraction 355/113.
What can a student take from al-Kashi?
Three habits stand out. He set a clear target before computing. He used a method he could repeat mechanically, so the work was long but not mysterious. And he carried extra precision through every step, because he knew where errors creep in. If the trigonometry or square roots behind the method feel shaky, the Fermi Concept Library covers every concept in mathematics, each with its own lesson and its place in the map.
Try it tonight: run the doubling formula from the hexagon to the 48-gon on a calculator, keeping six decimal places, and compare your answer with 3.141593.
