The Shulba Sutras are ancient Indian texts containing geometrical constructions and are considered the earliest sources on Indian mathematics. Four major Shulba Sutras were composed between 800 BCE to 200 CE, with the oldest attributed to Baudhayana around 800 BCE. The Baudhayana Shulba Sutra contains geometric constructions of shapes like squares and rectangles as well as area-preserving transformations. It provides what is considered one of the earliest statements of the Pythagorean theorem and an approximation of pi. The text also describes how to dissect a rectangle into three pieces that can be rearranged to form a square.

1. India’s Contribution to Geometry
Shulba Sutras (Geometrical constructions)
By
Dr. Bina Das
L.A.D. College for Women, Nagpur

2. GEOMETRICAL CONSTRUCTIONS
• The Shulba Sutras are part of the larger corpus
of texts called the Shrauta Sutras considered
to be appendices to the Vedas.
• They are the only sources of knowledge of
Indian Mathematics from the Vedic period.

3. • The four major Shulba Sutras, which are
mathematically the most significant, are those
composed by Baudhayana, Manava,
Apastamba and Katyayana, about whom very
little is known.
• The text have been dated from around 800
BCE to 200 CE, with the oldest being the sutra
that was attributed to Baudhayana around
800 BCE to 600 BCE.

4. • According to the theory of the ritual origins of
geometry, different shapes symbolized different
religious ideas.
• The Baudhayana Shulba sutra gives the
construction of geometric shapes such as squares
and rectangles. It also gives, sometimes approximate,
geometric area-preserving transformations from one
geometric shape to another.

5. Baudhāyana, (800 BC), a great hindu mathematician and
author of Baudhāyana Śulbasûtra that gave life to geometry

6. Baudhayana
• Baudhayana(800 BCE) was an Indian Mathematician,
who was most likely also a priest. He is noted as the
author of the earliest Shulba Sutra—appendices to
the Vedas giving rules for the construction of altars—
called the Baudhayana Sulbasutra, which contained
several important mathematical results. He is older
than the other famous mathematician Apastamba.
He belongs to the Yajurveda school.
• He is accredited with calculating the value of ‘pi'
before Pythagoras, and with discovering what is
known as the Pythagorean theorem.

7. In Baudhayana the rules are given as follows:
(i) The diagonal of a square produces double
the area (of the square).
(ii) The areas (of the squares) produced
separately by the lengths of the breadth of a
rectangle together equal the area (of the
square) produced by the diagonal.
(iii) This is observed in rectangles having sides 3
and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35,
15 and 36.

8. Squaring the circle
Squaring the circle: the areas of this square and this circle are equal.
In 1882, it was proven that this figure cannot be constructed in a finite
number of steps with an idealized compass and straightedge.

9. Squaring the circle is a problem proposed
by ancient geometers. It is the challenge of constructing
a square with the same area as a given circle by using only a
finite number of steps with compass and straightedge.
In 1882, the task was proven to be impossible, as a
consequence of the Lindemann–Weierstrass theorem which
proves that pi (π) is a transcendental, rather than an algebraic
irrational number; that is, it is not the root of
any polynomial with rational coefficients.

11. So the area
Now
Half-diagonal of the square is larger than the half-side by
radius = or , which equals
O

12. Area of the circle = 1.019 a2
= a2 + (.019) a2
Thus the area of circle is slightly larger than that of square by
a2(.019). The reason of this inequality is that the value of π is
irrational (and not exact).
Baudhayana gives a better result. He says: if sides of square is ‘a’ and
diameter of approximate circle is ‘d’ then
How he has obtained this result is a surprise?

13. A K L B
D M E C
H F G
Conversion of a rectangle into a square.
ABCD is a rectangle , AKMD is a square
ALGH is a required square

14. Turning a rectangle into a square by dissection
Consider a rectangle with sides a and b, such that,
a < b < 2*a. We show here one way to cut it into
three pieces and rearrange them into a square.

16. • Compute, s = √(a*b). Open a compass to length s, put a pin
in corner A of the rectangle, and mark point X on side CD.
The distance AX is s. Draw the line AX. Draw another line
(using an index card) BY, perpendicular to AX.
• Cut out the rectangle ABDC, cut the line AX, and cut the
line YB.

17. • In order to get a square in slanted position, move the left
triangle to the right, and move the top triangle to the
bottom right.

18. References
1. Baudhayana, wiki pedia the free encyclopedia.
2. Shulbha sutras, wiki pedia the free encyclopedia.
3. Squaring the circle, wikipedia the free encyclopedia.
4.Turning a rectangle into a square by dissection
http://sofia.nmsu.edu/~breakingaway/Lessons/R2S/R2S.html