Pappus’s centroid theorem

Pappus’s centroid theorem

Theorem 1

.

The

surface of revolution

generated by

a smooth curve

γ

in the xz-plane (with

x

≥

0

), rotated about the z axis,
has

surface area

A

=

s

⁢

d

,

where

s

is the

arc length

of

γ

, and

d

is the

distance

travelled
by the

centroid

μ

of

γ

under a full

rotation

.
(The centroid is also called the centre of mass, assuming the curve
has uniform line density.)

Theorem 2

.

The

solid of revolution

generated by a region (Lebesgue-measurable set)

S

in the xz-plane (with

x

≥

0

)
rotated about the z axis,
has volume

V

=

A

⁢

d

,

where

A

is the area of

S

, and

d

is the distance travelled by the centroid

μ

of

S

under a full rotation.

In English-speaking countries, these two theorems are known as Pappus’s theorems,
after the ancient Greek geometer Pappus of Alexandria.
In continental Europe, these theorems are more commonly associated with the name of
Paul Guldin (who rediscovered them):
e.g. in German “die guldinsche Regeln”; in Finnish “Guldinin säännöt”;
in French “le théorème de Guldin”.

Example 1

.

The surface area of the torus,
with the generating circle having radius

r

, and ring “radius”

R

(measured from the centre of the torus to the centre of the generating circle),
is

A

=

(

2

⁢

π

⁢

r

)

⁢

(

2

⁢

π

⁢

R

)

=

4

⁢

π

2

⁢

r

⁢

R

.
We used here the obvious fact that the centroid of a circle is at its centre.

Example 2

.

The volume of the (solid) torus, with the same parameters as above, is

V

=

(

π

⁢

r

2

)

⁢

(

2

⁢

π

⁢

R

)

=

2

⁢

π

2

⁢

r

2

⁢

R

.

Example 3

.

We compute the volume of the three-dimensional ball in

R

3

.
The ball can be considered to be the solid of revolution generated
by a half-disk.
So we will need to know the centroid of the upper-half disk

B

+

2

⁢

(

r

)

, radius

r

, in the plane.
By

symmetry

, this centroid has only a vertical component and no horizontal component.
The vertical component is calculated by:

1

π

⁢

r

2

/

2

⁢

∫

B

+

2

⁢

(

r

)

y

⁢

𝑑

x

⁢

𝑑

y

=

r

π

/

2

⁢

∫

B

+

2

⁢

(

1

)

y

⁢

𝑑

x

⁢

𝑑

y

=

r

π

/

2

⁢

∫

0

1

∫

-

1

-

y

2

1

-

y

2

y

⁢

𝑑

x

⁢

𝑑

y

=

r

π

/

2

⁢

∫

0

1

2

⁢

y

⁢

1

-

y

2

⁢

𝑑

y

=

4

⁢

r

3

⁢

π

.

Then the volume of the ball of radius

r

is

V

=

(

4

⁢

r

3

⁢

π

⋅

2

⁢

π

)

⁢

(

π

⁢

r

2

2

)

=

4

⁢

π

3

⁢

r

3

.

Example 4

.

Similarly we can compute the surface area of the sphere of radius

r

,
generated by revolving a half-circular arc.
The centroid of the upper half-circle

S

+

1

⁢

(

r

)

in the plane only has
the vertical component:

1

π

⁢

r

⁢

∫

0

π

y

⁢

r

⁢

𝑑

t

=

1

π

⁢

r

⁢

∫

0

π

(

r

⁢

sin

⁡

t

)

⁢

r

⁢

𝑑

t

=

r

π

⁢

∫

0

π

sin

⁡

t

⁢

d

⁢

t

=

2

⁢

r

π

.

Thus the surface area of the sphere is given by

A

=

(

2

⁢

r

π

⋅

2

⁢

π

)

⁢

π

⁢

r

=

4

⁢

π

⁢

r

2

.

Title

Pappus’s centroid theorem

Canonical name

PappussCentroidTheorem

Date of creation

2013-03-22 15:28:06

Last modified on

2013-03-22 15:28:06

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

7

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 53A05

Synonym

Guldin’s rule

Synonym

Guldinus theorem

Synonym

Guldin’s theorem

Synonym

Pappus’s theorem for solids of revolution

Synonym

Pappus’s theorem for surfaces of revolution

Related topic

Centroid2

Related topic

CentreOfMass

Related topic

SurfaceOfRevolution2

Related topic

VolumeOfSolidOfRevolution

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