Wednesday, August 26, 2015

Unizor - Geometry3D - Cavalieri's Principle





Unizor - Creative Minds through Art of Mathematics - Math4Teens

The
purpose of this lecture is to introduce the Cavalieri's principle that
will be used to derive the formula for a volume of a pyramid, the next
topic of this course.

The problem with a volume of a pyramid, as
with many other measures of geometric objects is that absolutely
rigorous derivation of the formulas for their area and volume can be
obtained only within a framework of higher levels of mathematics,
usually addressed in colleges. However, using the theory of limits on an
intuitive level, as presented in this course in a topic Algebra -
Limits is sufficient to derive these formulas for high school students.

The
Cavalieri's principle, actually, hides the complexity of this issue by
postulating certain property of geometric objects. This principle is not
an axiom mathematicians accept, it is a theorem that can be proven with
above mentioned higher levels of mathematics using integration.
However, accepting this principle as an axiom at this stage allows to
shortcut the derivation of formulas for area and volume of geometric
objects. The Cavalieri's principle sounds very natural and is
intuitively obvious. So, we will use it without any hesitation.

There
are two related to each other parts of the Cavalieri's principle - one
for area of a geometric object on a plane and another for volume of
solid objects in three-dimensional space.

Two-dimensional case

Assume we have two geometric objects X and Y on a plane.
Assume
also that there is a base line d on this plane such that any line h
parallel to this base line has the following property:
its
intersection with object X (a segment or a set of segments for
irregularly shaped object X) has the same linear measure as its
intersection with object Y.
Then the areas of objects X and Y are equal.

This
principle is a sufficient condition for equality of the areas of
objects X and Y, but by no means a necessary one. There are objects of
equal areas for which this principle is not true. For instance, take a
2x2 square and 1x4 rectangle. Their areas are equal, but you cannot find
a base line d such that any line parallel to it produces sections of X
and Y of the same linear measure.

Three-dimensional case

Assume we have two solid geometric objects X and Y in a three-dimensional space.
Assume
also that there is a base plane δ on this plane such that any plane γ
parallel to this base plane has the following property:
its intersection with object X (a flat object) has the same area as its intersection with object Y.
Then the volumes of objects X and Y are equal.

This
principle is a sufficient condition for equality of the volumes of
objects X and Y, but by no means a necessary one. There are objects of
equal volumes for which this principle is not true. For instance, take a
2x2x2 cube and 1x4x2 right rectangular prism. Their volumes are equal,
but you cannot find a base plane δ such that any plane parallel to it
produces sections of X and Y of the same area.

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