Archimedes Spher Cilindre-project

7/25/2019 Archimedes Spher Cilindre-project

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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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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

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Archimedes Spher Cilindre-project