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Builders in Vedic India had to make altars of exact shapes and areas using only ropes and pegs. Their rules were written down in texts called Sulba Sutras, and the oldest that survives is credited to Baudhayana. This post looks at what those rules say, how to check them, and where the topic sits in the NCERT books.
What is the Baudhayana Sulba Sutra?
The MacTutor biography of Baudhayana, from the University of St Andrews, says plainly that almost nothing is known about him except that he wrote one of the earliest Sulba Sutras. He was probably a Vedic priest and a skilled craftsman, and the text exists to give rules for building altars, not to do mathematics for its own sake. MacTutor's article on the Indian Sulbasutras dates the Baudhayana text to about 800 BC, with the Apastamba text about 600 BC and the Katyayana text about 200 BC. These dates are rough, and MacTutor notes that the texts give rules without proofs.
What does the rule of the diagonal say?
The new NCERT Class 8 maths book, Ganita Prakash Part II, Chapter 2 "The Baudhāyana-Pythagoras Theorem", starts with Baudhayana's question: how do you build a square with double the area of a given one? Doubling the side gives four times the area, not two. NCERT quotes his answer from verse 1.9: the diagonal of a square produces a square of double the area of the original square.
NCERT then quotes verse 1.12 for two different squares, "the area of the square produced by the diagonal is the sum of the areas of the squares produced by the two sides", and calls Baudhayana the first person to state the theorem in this general form. Historians do not all read the verses the same way. MacTutor's translation says the Baudhayana text gives only the square case explicitly, and credits the general rectangle version to the later Katyayana text. Both sources agree on what the rule is; they differ on who stated it in full first.
The Pythagoras side of the story, including the Babylonian evidence and Euclid's proof, is in our post on Pythagoras and the theorem.
Worked example 1: a square equal to a rectangle
MacTutor sets out the Baudhayana Sulba Sutra's construction for turning a rectangle into a square of the same area. Here it is with numbers, for a rectangle 4 units wide and 9 long (area 36).
Cut off a square. Mark a 4 × 4 square at one end. A 4 × 5 strip is left.
Halve the strip. Cut it into two 4 × 2.5 strips.
Move one strip. Put one along the side of the square and one along the end. You now have an L-shape inside a larger square of side 4 + 2.5 = 6.5, with a small 2.5 × 2.5 corner missing.
Take away the corner. The area is 6.5² − 2.5² = 42.25 − 6.25 = 36.
Find the side with a rope. The text uses a rope of length 6.5 swung round to meet a side, which gives the side of the required square as √(6.5² − 2.5²) = √36 = 6.
The answer, a 6 × 6 square, has area 36, the same as the rectangle. MacTutor checks the construction with the diagonal rule, since the last step is a right triangle with hypotenuse 6.5 and one side 2.5.
Worked example 2: squaring a circle, and how close it gets
MacTutor notes that all the Sulba Sutras give an approximate method for making a square equal in area to a circle: take the side as 13/15 of the diameter. Check it with a circle of diameter 30.
Square. Side = 13/15 × 30 = 26, so area = 676.
True circle. Area = π × 15² ≈ 706.86.
Error. 676/706.86 ≈ 0.956, so the square is about 4.4% too small.
Implied π. 4 × (13/15)² = 676/225 ≈ 3.004.
MacTutor lists other values implied by Baudhayana's constructions: 900/289 ≈ 3.114 and 1156/361 ≈ 3.202. The texts do not separate exact methods from approximate ones.
How good is the square root of 2?
MacTutor records that chapter 1, verse 61 of the Baudhayana Sulba Sutra gives √2 as 1 + 1/3 + 1/(3 × 4) − 1/(3 × 4 × 34) = 577/408. Watch the error shrink with each term (√2 = 1.41421356...):
1: too small by 0.414.
1 + 1/3 = 1.3333: too small by 0.081.
1 + 1/3 + 1/12 = 1.41667: too large by 0.0025.
577/408 = 1.4142157: too large by about 0.000002.
Our Pythagoras post squares 577/408 to show how close it is. Here is a different modern check: start from 17/12 and average it with 2 ÷ (17/12) = 24/17. The average is (289 + 288)/408 = 577/408. That is a modern way to see why the value is good. MacTutor does not say Baudhayana found it this way.
Common mistakes with these constructions
Doubling the side to double the area. That gives four times the area. Use the diagonal.
Adding sides instead of squares. Two squares of sides 3 and 4 combine into a square of side 5, not 7.
Treating approximations as exact. The circle construction is close, not exact.
The chapter is on the current NCERT Class 8 syllabus; check with your school for the year's plan. The square-root work connects to our guide on squares and square roots for Class 8. If you want a step-by-step check on a construction problem, the Fermi Problem Library has a solver that explains the reasoning rather than printing an answer.
What can a student take from the Sulba Sutras?
Test the rule with numbers. Every construction here can be checked by arithmetic.
Read sources carefully. NCERT and MacTutor agree on the mathematics and differ on who said what first.
Practice routine: turn a 2 × 8 rectangle into a square using the steps above, combine squares of sides 5 and 12, and work out the error in the 13/15 rule for a circle of diameter 15.
Next step: take a piece of string, lay it across the diagonal of a square drawn on the floor, and build the double square.
