In general we can calculate the area of any
POLYGON
as the sum of the areas of the triangles that subdivide
it. For example, the area of a
TRAPEZOID
is the sum
of the areas of two triangles, and the area of a regu-
lar
POLYGON
with nsides is the sum of the areas of n
triangles.
Curved Figures
It is also possible to compute the area of curved fig-
ures. For example, slicing a circle into wedged-shape
pieces and rearranging these slices, we see that the
area of a circle is close to being the area of a rectangle
of length half the circumference and of width r, the
radius of the circle.
If we work with finer and finer wedged-shape
pieces, the approximation will better approach that of
a true rectangle. We conclude that the area of a circle is
indeed that of this ideal rectangle:
area of a circle = 1/2×circumference ×r
(Compare this with the formula for the area of a
regular polygon.) As
PI
(π) is defined as the ratio
of the circumference of a circle to its diameter,
, the area of a circle can thus be
written: area = 1/2×2πr×r. This leads to the famous
formula:
area of a circle = πr2
The methods of
INTEGRAL CALCULUS
allow us to
compute areas of other curved shapes. The approach is
analogous: approximate the shape as a union of rectan-
gles, sum the areas of the rectangular pieces, and take
the
LIMIT
of the answers obtained as you work with
finer and finer approximations.
Theoretical Difficulties
Starting with the principle that a fundamental shape, in
our case a rectangle, is asserted to have “area” given by
a certain formula, a general theory of area for other geo-
metric shapes follows. One can apply such an approach
to develop a measure theory for measuring the size of
other sets of objects, such as the notion of the surface
area of three-dimensional solids, or a theory of
VOLUME
.
One can also develop a number of exotic applications.
Although our definition for the area of a rectangle
is motivated by intuition, the formula we developed is,
in some sense, arbitrary. Defining the area of a rectan-
gle as given by a different formula could indeed yield a
different, but consistent, theory of area.
In 1924 S
TEFAN
B
ANACH
and Alfred Tarski
stunned the mathematical community by presenting a
mathematically sound proof of the following assertion:
It is theoretically possible to cut a solid ball
into nine pieces, and by reassembling them,
without ever stretching or warping the pieces,
form TWO solid balls, each exactly the same
size and shape as the original.
This result is known as the Banach-Tarski paradox, and
its statement—proven as a mathematical fact—is abhor-
rent to our understanding of how area and volume
should behave: the volume of a finite quantity of material
should not double after rearranging its pieces! That our
intuitive understanding of area should eventually lead to
such a perturbing result was considered very disturbing.
What mathematicians have come to realize is that
“area” is not a well-defined concept: not every shape in
a plane can be assigned an area (nor can every solid in
three-dimensional space be assigned a volume). There
exist certain nonmeasurable sets about which speaking
of their area is meaningless. The nine pieces used in the
Banach-Tarski paradox turn out to be such nonmeasur-
able sets, and so speaking of their volume is invalid.
(They are extremely jagged shapes,
FRACTAL
in nature,
and impossible to physically cut out.) In particular,
interpreting the final construct as “two balls of equal
volume” is not allowed.
Our simple intuitive understanding of area works
well in all practical applications. The material pre-
sented in a typical high-school and college curriculum,
for example, is sound. However, the Banach-Tarski
paradox points out that extreme care must be taken
circumference
π= ––––––––––––
–
2r
24 area
Establishing the area of a circle