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Archimedes :
A Sphere and A Cylinder
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Archimedes said: The surface area of a sphere equales the surface
area of a cylinder whose height from the base to the top and
the diameter of the base are the same as the diamiter of the sphere.
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Before going to the details of what Archimedes did,
we first must review how to calculate the area of the
conical frustum.
First, let us consider the area of a cone.
The Area of Cone
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As we know, we can cut a cone
and make it 2D.
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Area of Cone = (1/2) x 2(pi)m x s
s
2(pi)m
s
---------------m---------------
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Now,consider the area of a conical frustum.
from the side view of a cone, we get
t:m = (t+s) : M
tM = (t+s)m
t(M-m)=sm
M
H
m
h t
s
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Top View of The Area of
The Conical Frusrum in 2D
t s
2(Pi)m
2(pi)M
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Area = (1/2)x(2(pi)Mx(s+t))
Area = (1/2)x(2(pi)mxt)
The Area of above Conicol Frustum= {(1/2)x(2(pi)Mx(s+t))}- {(1/2)x(2(pi)mxt)}
=(pi)M x (s+t)-(pi)m x t = (pi)(Ms+Mt-mt)
From the side vies of Conicol Frustum: t(M-m) =sr-----(a)
substitute (a) in this equation,
Area of Conicol Frustum
=(pi)(Ms-ms)=(pi)(M-m)s
= -
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The surface area of a sphere:
When a polygon is inscribed inside the great circle
of a sphere and rotated around an axis,the surface area
of polyhedron gets very close to the surface area of
sphere as n gets larger and larger.
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The surface area of a sphere:
When a polygon is inscribed inside the great circle
of a sphere and rotated around an axis,the surface area
of polyhedron gets very close to the surface area of
sphere as n gets larger and larger.
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The surface area of a sphere:
When a polygon is inscribed inside the great circle
of a sphere and rotated around an axis,the surface area
of polyhedron gets very close to the surface area of
sphere as n gets larger and larger.
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The surface area of a sphere:
When a polygon is inscribed inside the great circle
of a sphere and rotated around an axis,the surface area
of polyhedron gets very close to the surface area of
sphere as n gets larger and larger.
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The surface area of a sphere:
When a polygon is inscribed inside the great circle
of a sphere and rotated around an axis,the surface area
of polyhedron gets very close to the surface area of
sphere as n gets larger and larger.
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The surface area of a sphere:
When a polygon is inscribed inside the great circle
of a sphere and rotated around an axis,the surface area
of polyhedron gets very close to the surface area of
sphere as n gets larger and larger.
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Archimedes method:
Use a cone to calculate
the area of a sphere
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Archimedes rotated the circle below
along the axis AD.
Since the triangle ABG
and the triangle ABD are
similar, m(=GB) : AG = Y : s.
so, ms = (AG)Y.Since the triangle AGB is
similar to the triangl HGP,
m : AG = m :Gp = Y :s.
so, ms = (GP)Y.
Similarly, M : PF = m : GP = Y : s.
So, Ms = (PF)Y.
Area of Conical Frustum (blue)
= pi(m+M)s
= pi( ms + Ms)
= pi{(GP)Y + (PF)Y}
= pi{(GP +PF)Y}= pi(GF)Y
As n goes to infinity, Ybecomes inc
to the diameter of the
great circle. Therefore,
Area of the Conical Frustum
= 2(pi)r(GF), where r is a radius
of the great circle of a sphere.
A
B
C
D
E F
GH
m
M
Y
s
P
M
H
m
h t
s
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The side view of a cylinder----------r---------
h
2(pi)r
hArea= 2(pi)rh