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S
P
H
E
R
E
• Circle is the only shape where there is only one
edge, no straight lines, and a curve that is
completely unified for a full 360 degrees around
a single center point. It resolves to One, and
thus it is the simplest possible two-dimensional
shape.
• When we expand this into three dimensions, we
can then see that the similar principle applies to
the sphere. Physicist Buckminister. Fuller
described a sphere as "a multiplicity of discrete
events, approximately equidistant in all
directions from a nuclear center."
Some interesting Facts :
 
• It is perfectly symmetrical
• It has no edges or vertices
(corners)
• It is not a polyhedron
• All points on the surface are
the same distance from the
center
Surface Area = 4 × π × r2
Volume = (4/3) × π × r3
Largest Volume for Smallest Surface
Of all the shapes, a sphere has the smallest surface
area for a volume. Or put another way it can contain the
greatest volume for a fixed surface area.
Example: if you blow up a balloon it naturally forms a
sphere because it is trying to hold as much air as possible with
as small a surface as possible.
In Nature
The sphere appears in nature whenever a surface
wants to be as small as possible. Examples include bubbles
and water drops, can you think of more?
The Earth
The Planet Earth,
our home, is nearly a
sphere, except that it is
squashed a little at the
poles.
It is a spheroid, which means it just misses
out on being a sphere because it isn't perfect in one
direction (in the Earth's case: North-South)
A sphere can be compressed into a single point,
which has no space and no time, and thus exist as the
simplest object in the Universe, but the sphere also is the
most complex form in the Universe, containing all other
things within itself.
In a sphere you can draw an infinite number of
lines that connect to an infinite number of points (i.e.
“events ”) on the surface of the sphere, with all the lines
starting from one single center point or nucleus, and all
the lines will come out to be the exact same length.
This makes the sphere the most complex three-
dimensional object that there is; an infinite number of
different geometric shapes can be drawn inside of it, by
simply connecting different points on the surface of the
sphere together.
Once you stretch or flatten the sphere in any way,
you have less symmetry and thus have less flexibility in
what can be geometrically created inside. (This may
seem hard to understand, but it can be proven
mathematically.
This also explains why liquid naturally forms into
spheres when it is in free-fall and/or in a soap bubble, as
the air pressure on the liquid is equal on all sides.) The
sphere is also the simplest three-dimensional formation
in the Universe for the same reasons as the circle;
namely, there is only one edge, perfectly symmetrical in
its curvature around a center point, and thus all resolves
to One.
ARYABHATTA–I (476 BC
– 550 AD) was 1st
to
include the computation of
areas & volumes of sphere
in his literature.
ARYABHATTA – I
Aryabhatta gives a wrong formula for the
volume of a sphere.
leifj.kkgL;k?kZ fo"dEHkk/kZgreso o`RrQye~A
rfUutewysu gra ?kuxksyQy fujo'ks"ke~AA
(A. B. Ganitapada. 7)
(Half the circumference multiplied by half the
diameter is the area of a circle. That multiplied by its
own root is the exact volume of a sphere.
According to this the volume of sphere =
8
d.
4
d
.
4
d 22/322
π
=
ππ
=
Bhaskara and the later mathematicians give
correct formulae for both the surface area and the
volume of the sphere.
o`r{ks=s ifjf/kxqf.krO;klikn% Qy ;&
R{kqi.ka osnS:ifjifjr% dUnqdL;so tkye~A
xksyL;Soa rnfi p Qya i`"Bta O;klfu?ue~
"kM~fHkHkZDra Hkofr fu;ra xksyxHksZ ?
kuk[;e~AA
(Lil. 201)
(In a Circle, the circumference multiplied by one-
fourth the diameter is the area, which, multiplied by 4,
is its surface area going round it like a net round a
ball. This (surface area) multiplied by the diameter
and divided by 6 is the volume inside the sphere.)
i.e. area of a circle =
= circumference .
The surface area of a sphere = 4. area of its great circle
= 4πr2
Volume of a sphere =
= Surface area .
2
2
r
4
d
4
d π=
π
=
3
r
3
4
6
r2 π=
In modern mathematics
the formula of volume of
sphere can be derived using
following method i.e. disc
integration to sum the
volumes of an infinite no. of
circular discs of infinitesimally
small thickness stacked
centered side by side along
the axis from x = 0 where the
disc has radius r
i.e. y = r to x = r
At any given x the incremental volume
(δv) is given by the product of cross sectional area of
the disc at x and its thickness (δx)
δv ≈πy2
δx
The total volume is summation of all
incremental volumes
v ≈Σπy2
δx
in the limit as δx approaches zero this
becomes
∫−
π=
r
r
2
dxyv
At any given x a right angle triangle connects x, y and r
to the origin, hence it follows from the geometry that y2
= r2
–
x2
Thus substituting y with a function of x gives
This can be now evaluated as
∫−
−π=
r
r
22
dx)xr(v
3
r3
2
r
3
4
3
x
xrv π=





−π=
-r
Reference :-
• Balachandra Rao, S. Indian Mathematics and
Astronomy. Jnana Deep Publications. 1994
• Barathi Krishna Tirtha. Vedic Mathematics. Motilal
Banrsidas. 1992
• Bhanu Murthy. T.S. A Modern Introduction to Ancient
Indian Mathematics. Wiley Eastern. 1992
4. Durant, Will. The Story of Civilization. Part I: Our
Oriental Heritage. Simon and Schuster, New York,
1954.
Volume of sphere[1]

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Volume of sphere[1]

  • 1.
  • 3. • Circle is the only shape where there is only one edge, no straight lines, and a curve that is completely unified for a full 360 degrees around a single center point. It resolves to One, and thus it is the simplest possible two-dimensional shape. • When we expand this into three dimensions, we can then see that the similar principle applies to the sphere. Physicist Buckminister. Fuller described a sphere as "a multiplicity of discrete events, approximately equidistant in all directions from a nuclear center."
  • 4. Some interesting Facts :   • It is perfectly symmetrical • It has no edges or vertices (corners) • It is not a polyhedron • All points on the surface are the same distance from the center Surface Area = 4 × π × r2 Volume = (4/3) × π × r3
  • 5. Largest Volume for Smallest Surface Of all the shapes, a sphere has the smallest surface area for a volume. Or put another way it can contain the greatest volume for a fixed surface area. Example: if you blow up a balloon it naturally forms a sphere because it is trying to hold as much air as possible with as small a surface as possible. In Nature The sphere appears in nature whenever a surface wants to be as small as possible. Examples include bubbles and water drops, can you think of more?
  • 6. The Earth The Planet Earth, our home, is nearly a sphere, except that it is squashed a little at the poles. It is a spheroid, which means it just misses out on being a sphere because it isn't perfect in one direction (in the Earth's case: North-South)
  • 7. A sphere can be compressed into a single point, which has no space and no time, and thus exist as the simplest object in the Universe, but the sphere also is the most complex form in the Universe, containing all other things within itself. In a sphere you can draw an infinite number of lines that connect to an infinite number of points (i.e. “events ”) on the surface of the sphere, with all the lines starting from one single center point or nucleus, and all the lines will come out to be the exact same length. This makes the sphere the most complex three- dimensional object that there is; an infinite number of different geometric shapes can be drawn inside of it, by simply connecting different points on the surface of the sphere together.
  • 8. Once you stretch or flatten the sphere in any way, you have less symmetry and thus have less flexibility in what can be geometrically created inside. (This may seem hard to understand, but it can be proven mathematically. This also explains why liquid naturally forms into spheres when it is in free-fall and/or in a soap bubble, as the air pressure on the liquid is equal on all sides.) The sphere is also the simplest three-dimensional formation in the Universe for the same reasons as the circle; namely, there is only one edge, perfectly symmetrical in its curvature around a center point, and thus all resolves to One.
  • 9. ARYABHATTA–I (476 BC – 550 AD) was 1st to include the computation of areas & volumes of sphere in his literature. ARYABHATTA – I
  • 10. Aryabhatta gives a wrong formula for the volume of a sphere. leifj.kkgL;k?kZ fo"dEHkk/kZgreso o`RrQye~A rfUutewysu gra ?kuxksyQy fujo'ks"ke~AA (A. B. Ganitapada. 7) (Half the circumference multiplied by half the diameter is the area of a circle. That multiplied by its own root is the exact volume of a sphere. According to this the volume of sphere = 8 d. 4 d . 4 d 22/322 π = ππ =
  • 11. Bhaskara and the later mathematicians give correct formulae for both the surface area and the volume of the sphere. o`r{ks=s ifjf/kxqf.krO;klikn% Qy ;& R{kqi.ka osnS:ifjifjr% dUnqdL;so tkye~A xksyL;Soa rnfi p Qya i`"Bta O;klfu?ue~ "kM~fHkHkZDra Hkofr fu;ra xksyxHksZ ? kuk[;e~AA (Lil. 201) (In a Circle, the circumference multiplied by one- fourth the diameter is the area, which, multiplied by 4, is its surface area going round it like a net round a ball. This (surface area) multiplied by the diameter and divided by 6 is the volume inside the sphere.)
  • 12. i.e. area of a circle = = circumference . The surface area of a sphere = 4. area of its great circle = 4πr2 Volume of a sphere = = Surface area . 2 2 r 4 d 4 d π= π = 3 r 3 4 6 r2 π=
  • 13. In modern mathematics the formula of volume of sphere can be derived using following method i.e. disc integration to sum the volumes of an infinite no. of circular discs of infinitesimally small thickness stacked centered side by side along the axis from x = 0 where the disc has radius r i.e. y = r to x = r
  • 14. At any given x the incremental volume (δv) is given by the product of cross sectional area of the disc at x and its thickness (δx) δv ≈πy2 δx The total volume is summation of all incremental volumes v ≈Σπy2 δx in the limit as δx approaches zero this becomes ∫− π= r r 2 dxyv
  • 15. At any given x a right angle triangle connects x, y and r to the origin, hence it follows from the geometry that y2 = r2 – x2 Thus substituting y with a function of x gives This can be now evaluated as ∫− −π= r r 22 dx)xr(v 3 r3 2 r 3 4 3 x xrv π=      −π= -r
  • 16. Reference :- • Balachandra Rao, S. Indian Mathematics and Astronomy. Jnana Deep Publications. 1994 • Barathi Krishna Tirtha. Vedic Mathematics. Motilal Banrsidas. 1992 • Bhanu Murthy. T.S. A Modern Introduction to Ancient Indian Mathematics. Wiley Eastern. 1992 4. Durant, Will. The Story of Civilization. Part I: Our Oriental Heritage. Simon and Schuster, New York, 1954.