1. Sridharacharya’s contribution
to Geometry
RADHIKA YELKAWAR
ASSOCIATE PROFESSOR
DEPARTMENT OF MATHEMATICS
L.A.D. COLLEGE
NAGPUR

2. SRIDHARACHARYA
(c. 870– c. 930)
Sridharacharya is believed to have lived
between seventh to eleventh century.
The best present estimate is 900 AD
Birth place- Hooghly District in West
Bengal or South India
Father's name - Baladevacharya
Mother's name- Acchoka.

3. Works
Sridhara is known as the author of two mathematical
treatises, namely
Patiganita
Trisatika
However at least three other works have been attributed to
him, namely
Bijaganita
Navasati (having nine hundred)
Brhatpati (bigger pati)
Information about these books was given in the works of
Bhaskara II (around 1150), Makkibhatta (in 1377) and
Raghavabhatta (in 1493).

4. Works (contd……)
 Of all the Hindu Acharyas, the description of Sridharacharya on zero is
the most explicit. He has written,
 If 0 (zero) is added to any number, the sum is the same number.
 If 0 (zero) is subtracted from any number, the number remains unchanged.
 If 0 (zero) is multiplied by any number, the product is 0(zero).
 He has said nothing about division of any number by 0(zero).
 In the case of dividing a fraction, he has found out the method of
multiplying the fraction by the reciprocal of the divisor
 Wrote on practical applications of algebra and also separated algebra
from arithmetic’s
 One of the first to give a formula for solving quadratic equations.

5. Patiganita
Patiganita is the most important work of Sridhara.
Throughout the book, Sridhara has given methods to solve
problems in terse rules in the form of verses
No proofs are given..
 The book is divided into four parts
 It contains series of problems, some of which are only
approximate.
 Only one copy of this book is survived and its last pages
are missing

6. Patiganita
 First Part
• money measure
• weights
• measure of capacity
• linear measure
• time measurement

7. Patiganita
 Second part (Prakarama)
• Addition, subtraction, multiplication, division,
• squaring and square root,
• cubing and cube root
• operations for fractions, reduction of fractions
• rule of three, inverse rule of three, rules of five,
seven and nine
• barter of commodities, sale of living beings etc.

8. Patiganita
Third part
 simple interest
 valuation of pieces of gold
 partnership
 purchase and sale
 wages and payments, wages paid from the commodity
 series in arithmetic progression, series in geometric
progressions
 miscellaneous problems on series in arithmetic
progression
 series of squares and cubes of the terms of an
arithmetic series.

9. Patiganita
 Fourth part
 Area of quadrilateral with equal and unequal
altitudes
 Area of the triangle
 Area and circumference of circle
 Area of segment of circle
 Surface area and volume of a sphere

10. Trisatika
 This work of Sridhara is also known as Patiganitasara
 Patiganitasara is a summary of the Patiganita including the missing
portion.
 It is called Trisatika as it contains 300 verses.

11. Volume of sphere
 Construction of perfect domes in the ancient structures is a testimony
of knowledge of our ancient civilizations that there is a fixed
relationship (𝝅 ) between circumference of a circle and its diameter.
 The value of 𝜋 =
𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒
𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒
 Formula for volume of the sphere i.e. 𝑉 =
4
3
𝜋𝑟3 was first derived by
Archimedes. This formula uses a symbol 𝜋.
 𝜋 is an irrational number and therefore there is no exact value of 𝜋.
The value of 𝜋 now we know is ≈
22
7
≈3.14

12. V =
𝟒𝛑
𝟑
𝐫 𝟑
=
𝟒
𝟑
×
𝟐𝟐
𝟕
𝑟3
=
88
21
𝑟3
≈ 4.19 𝑟3
-----(1)
Volume of sphere
Archimedes formula

13. Meaning:Half the cube of the diameter of a sphere combined with its
18th part is the volume of a sphere
i.e. If d is the diameter of the sphere then
𝐕 =
𝐝 𝟑
𝟐
+
𝟏
𝟏𝟖
(
𝐝 𝟑
𝟐
)
=
19
36
𝑑3
=
19
36
(2r)3
where d = 2r, r = radius of a sphereType equation here.
=
38
9
𝑟3
≈ (𝟒. 𝟐𝟐)𝐫 𝟑
-----------(2)
Volume of sphere

14. Value of 𝜋
Thus the volume of sphere computed by Sridharacharya is very close to
the value computed by Archimedes and it differs only with the value of 𝝅.
From (1) and (2)
38
9
𝑟3
=
4𝜋
3
𝑟3

38
9
=
4𝜋
3
𝜋 =
38
9
×
3
4
 𝜋 =
19
6
≅ √10
Thus It appears that Sridharacharya considered the value of 𝝅 = √𝟏𝟎

15. Area and Circumference of a
circle
Sridhara has computed value of 𝜋 in the following verse
Meaning of first line- The circumference of a circle is equal
to the square root of 10 times the square of its diameter
i.e. Circumference of circle = 10𝑑2 = 10 × 𝑑
Meaning of second line- The area of a circle is the square root
of the product of 10 with the square of semi-diameter square
Area of circle = 10
𝑑
2
2 2
= 10 ×
𝑑
2
2

16. Area of segment of a circle
Meaning : The square of the arrow as multiplied by half the sum of the chord
and the arrow should be multiplied by 10 and divided by 9. The square root
of the quotient gives the area of the segment of a circle
Area of a segment of a circle = ℎ ×
𝑐+ℎ
2
2
×
10
9
=
𝜋
3
ℎ ×
𝑐+ℎ
2
2