International Journal of Engineering Trends and Technology (IJETT) – Volume 27 Number 1- September 2015
A note on Simple but Interesting Mathematical results as
Inculsive Technology for Sustainable Solutions
Dr. N.Shivakumar
Dr.S.Sridhar
Professor & Head
Department of Mathematics
R.V. College of Engineering,
Mysore road, Bangalore-560069, India
Professor & Dean
Cognitive & Central Computing
R.V.College of Engineering,
Mysore road, Bangalore-560069, India
Abstract
In this paper we present some simple but interesting Mathematics of importance. The results are :- A) the
circumference of any right circular cylinder is obtained by multiplying the square of its diameter by 10 and then
taking the square root of the product. When the circumference of such a cylinder is multiplied by 1/4 th of its
diameter we get its area. This will be very useful as inclusive technology in cutting design of metal sheet without
wastage in optimum manner in fabrication industries whenever they deal with costly metal sheet like brass or
aluminum for manufacturing cone or cylinder type of containers for multiple purposes B)Formula for finding the
sum to ‘n’ terms of a series.. C) From the square of the sum of the arrow and the diameter is subtracted the square
of the diameter. The remainder is doubled, giving the square of the arc and its square root gives the measure of the
arc. This type of result will be useful in the field of civil engineering designs. Thus, these results are very useful for
day to day problems as Inculsive technology for sustainable solutions.
Key words: circular cylinder, diameter, circumference, series, arc, area.
Result1.
The circumference of any right circular cylinder is obtained by multiplying the square of its diameter by 10 and
then taking the square root of the product. When the circumference of such a cylinder is multiplied by
1
th of its
4
diameter we get its area.
Let us look at the following figure mathematically.
circumference of the top of the right circular cylinder will be
If d is the diameter of the base/top, then the
d 2  10  10 d. If d  2r and π  10 then
this formula gets reduced to circumference of the circle = 2 r , which is presently used.
A
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C
B
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International Journal of Engineering Trends and Technology (IJETT) – Volume 27 Number 1- September 2015
Again area of the top (circle) =
10  d 
d
10

 d2
4
4
If d  2r & 10   then this formula is  r 2
Surface area of the cylinder 
diameter 2  10  diameter
4
diameter
4
diameter
 10  diameter
Square yojanas 
4
Volume of the cylinder  area of the base  depth of pit
 circumference 
10
2
 diameter   depth cube yojanas 
4
2

1
19 1
 10
 1

4
6
4
19

cubic yojana s
24

Note:
Explanation of
10 
19
6
2
N = a r
N  a2  r
a
r
2a
10  32  1
 3

1
1
 3
23
6
19
6
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International Journal of Engineering Trends and Technology (IJETT) – Volume 27 Number 1- September 2015
Result2.
Formula for finding the sum to ‘n’ terms of a series.
Here the use of icchå is trivial, although one can place it in the formula to denote a particular earth, say (
e 1 , e 2 .......... e 6 )
The formula is given by
Ser = [{(number of terms - icchå) 
(common difference)}
+ {(icchå – 1)  common difference} + (first term  2)] 
no. of terms
2
where r = 1,2,3,4,5,6.
i.e.
Ser = [{ ( n
 e r ) d  ( e r  1 ) d  2 a }] 
n
2
where n = No. of terms, d = common difference, a = first term, e r = icchå,
r = 1 to 6
This is the same as the usual formula,
s 
n
[ 2 a  ( n  1) d ]
2
In order to find out the measure of the central and the sequence-ordered holes of all the earth’s, 5 is the first
term, 8 is the common difference, and 49 is the number of terms. This is to be understood well.
The requisition in excess of the number of terms is multiplied by the common difference; out of the
product, the requisition in excess of unity as multiplied by common difference is subtracted. Whatever is the
remainder, to it is added twice the first term, and then it is multiplied by half of the number of terms. This gives the
sum of the series.
Let the number of terms =
n
Common difference
=
d
First term
=
a
Icchå
=
er
r
=
1,2....7
According to the above verse the formula is given by
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International Journal of Engineering Trends and Technology (IJETT) – Volume 27 Number 1- September 2015
S total 
7
 n
n 

 7


n

e

d

  e r  1d  2a   2a  n  1d 


r 
2 
1

 1

 2
Now it is given in the verse that n =49, d = 8, a = 5
Therefore S total 
49
2  5  49  18  9653 for all the seven earths.
2
Result3:
Measure of the arc
From the square of the sum of the arrow and the diameter is subtracted the square of the diameter. The
remainder is doubled, giving the square of the arc (dhanu¼a) and its square root gives the measure of the arc.
The above verses have the following mathematical information.
O
Chord
r-h
P
height of
segment
A
B
h
Q
The formula for finding out the chord of a circle when the height of the segment and diameter of the circle
are given.
Let D be the diameter, h be the height of the segement, 'r' be the radius; i.e.,
D
2
2
 D  2  D
 
According to the verse, the Chord  4.     h  
 
 2   2
This may also be written as Chord =
ISSN: 2231-5381

4 r 2  r  h 
2

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International Journal of Engineering Trends and Technology (IJETT) – Volume 27 Number 1- September 2015
Hence, by Pythagoras theorem
OB 2  OP 2  PB 2
PB 2  OB 2  OP 2
or
OB2  OP 2
PB 


2 PB  4 OB   OP  AB
2
 D 2  D  2h 
2
2
The second verse gives the formula for finding out the arc intercepted by the chord of a circle, when again
the height of the segment and the diameter of the circle are given. The formula for finding out the intercepted arc is

arc  2 D  h   D 2
2

The square of chord of the bow (arc) between the (two consecutive) cardinal points (in the form of onefourth of the circumference) of the circular island also is twice the square of half of the diameter. When this square
is multiplied by five and divided by four the square of the bow (arc) is obtained and its square root gives the
measure of the arc .The mathematics is as follows :
If the diameter is taken as D and the circumference as c, the radius as 'r', then
100000 yojana
B
C
Chord
quadrant
2
D
2
  2  2r
2
(chord of a quadrant arc)2 = 
arc
A
Alternatively,
Chord AB 2   AC 2  CB 2  2r 2
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International Journal of Engineering Trends and Technology (IJETT) – Volume 27 Number 1- September 2015
The relation between the fig. quadrant arc and the chord has been shown by the author as follows -
Chord of the quadrant arc2  5
(quadrant circumference)2=
4
=
2.
D 2 5 5D 2 10r 2
 

4 4
8
4
by taking square root quadrant circumference as 10
r
2
Conclusions
Thus we have seen some simple but interesting Mathematics of importance. The results are :-A) The circumference
of any right circular cylinder is obtained by multiplying the square of its diameter by 10 and then taking the square
root of the product. When the circumference of such a cylinder is multiplied by 1/4 th of its diameter we get its
area. This will be very useful as inclusive technology in cutting design of metal sheet without wastage in optimum
manner in fabrication industries whenever they deal with costly metal sheet like brass or aluminum for
manufacturing cone or cylinder type of containers for multiple purposes B)Formula for finding the sum to ‘n’ terms
of a series.. C) From the square of the sum of the arrow and the diameter is subtracted the square of the diameter.
The remainder is doubled, giving the square of the arc and its square root gives the measure of the arc. This type of
result will be useful in the field of civil engineering designs. Thus, these results are very useful for day to day
problems as Inculsive technology for sustainable solutions.
References
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7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
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