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.