![Pappus centroid theorem areas](https://upload.wikimedia.org/wikipedia/commons/f/fd/Pappus_centroid_theorem_areas.gif)
In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
s dealing with the
surface areas and
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
s of
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s and
solid
Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structural ...
s of revolution.
The theorems are attributed to
Pappus of Alexandria
Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
and
Paul Guldin
Paul Guldin (born Habakkuk Guldin; 12 June 1577 ( Mels) – 3 November 1643 (Graz)) was a Swiss Jesuit mathematician and astronomer. He discovered the Guldinus theorem to determine the surface and the volume of a solid of revolution. (This theor ...
. Pappus's statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640.
The first theorem
The first theorem states that the
surface area ''A'' of a
surface of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...
generated by rotating a
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
''C'' about an
axis
An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to:
Mathematics
* Axis of rotation: see rotation around a fixed axis
* Axis (mathematics), a designator for a Cartesian-coordinat ...
external to ''C'' and on the same plane is equal to the product of the
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
''s'' of ''C'' and the distance ''d'' traveled by the
geometric centroid of ''C'':
:
For example, the surface area of the
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
with minor
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
''r'' and major radius ''R'' is
:
Proof
A curve given by the positive function
is bounded by two points given by:
and
If
is an infinitesimal line element tangent to the curve, the length of the curve is given by:
The
component of the centroid of this curve is:
The area of the surface generated by rotating the curve around the x-axis is given by:
Using the last two equations to eliminate the integral we have:
The second theorem
The second theorem states that the
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
''V'' of a
solid of revolution generated by rotating a
plane figure
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie o ...
''F'' about an external axis is equal to the product of the area ''A'' of ''F'' and the distance ''d'' traveled by the geometric centroid of ''F''. (The centroid of ''F'' is usually different from the centroid of its boundary curve ''C''.) That is:
:
For example, the volume of the
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
with minor radius ''r'' and major radius ''R'' is
:
This special case was derived by
Johannes Kepler using infinitesimals.
Proof 1
The area bounded by the two functions:
where
where
and bounded by the two lines:
and
is given by:
The
component of the centroid of this area is given by:
If this area is rotated about the y-axis, the volume generated can be calculated using the shell method. It is given by:
Using the last two equations to eliminate the integral we have:
Proof 2
Let
be the area of
,
the solid of revolution of
, and
the volume of
. Suppose
starts in the
-plane and rotates around the
-axis. The distance of the centroid of
from the
-axis is its
-coordinate
:
and the theorem states that
:
To show this, let
be in the ''xz''-plane,
parametrized by
for
, a parameter region. Since
is essentially a mapping from
to
, the area of
is given by the
change of variables
Change or Changing may refer to:
Alteration
* Impermanence, a difference in a state of affairs at different points in time
* Menopause, also referred to as "the change", the permanent cessation of the menstrual period
* Metamorphosis, or change, ...
formula:
:
where
is the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the
Jacobian matrix of the change of variables.
The solid
has the
toroidal parametrization
for
in the parameter region