planar figures are in perspective from a point, then they
are also in perspective from a line, and vice versa.
(Although the theorem is stated only for triangles, any
polygonal figure can be subdivided into triangles for
which the theorem applies.)
pi The real number, denoted, π, defined as the ratio
of the circumference C of a CIRCLE of any size drawn in
a plane to its diameter D, is called pi:
Its value is π = 3.141592653589793… That the value of
this ratio is the same for all planar circles is not immediately obvious. To see why this is the case, note that if
any diagram or picture is scaled up or down by a factor
k, then all lengths in that picture change by a factor k. In
particular, for a picture of a circle, the length of its circumference changes to kC and the length of its diameter
to kD. Consequently, the ratio of circumference to
diameter remains unchanged:
=
. That any two
circles in the plane can be thought of as scaled versions
of each other thus explains why the value of π is the
same for all circles. (It is worth mentioning that the
property of scaling used here is a phenomenon of
EUCLIDEAN GEOMETRY and that the value of π is not the
same for all circles in SPHERICAL GEOMETRY or HYPERBOLIC GEOMETRY. For example, the value of π changes
for circles drawn on spheres, ranging in value anywhere
between 2 and 3.141592…)
A study of AREA shows that the area of a circle of
radius r is given by:
A = πr
2
Thus π can also be defined as the ratio of the area of a
circle to its radius squared.
Although it is not standard, one can associate a
value π to other shapes as well. For instance, the ratio of
the perimeter of a square to its width (short diameter) is
the same for all squares. Thus we may say: π square = 4.
(Using the diagonal of the square as the long diameter
produces instead the value π square =
≈ 2.828.) The
ratio of the perimeter of an equilateral triangle to its
diameter (measured as the height of the triangle) is the
same for all equilateral triangles. We have: π triangle =
≈ 3.464. In fact, in precisely this way, by inscribing
polygons with increasingly many sides within a circle
and calculating the associated values of π for these polygons, ARCHIMEDES OF SYRACUSE (ca. 287–212 B.C.E.)
showed that the true value of π for circles lies between
= 3 +
and
= 3 +
. The approximation
, correct to two decimal places, is still often used
today.
Since the time of antiquity, scholars have attempted
to find the exact value of π. The RHIND PAPYRUS from
ancient Egypt (dated ca.1650 B.C.E.) describes a procedure for computing the area of a circle that is
equivalent to using the approximate value
for π.
The Babylonians of the same era used the approximate
value
. Around 150 C.E. Greek astronomer CLAUDIUS
PTOLEMY established that π ≈
, and 300 years later
Chinese scholar ZU CHONGZHI used the improved
approximation π ≈
, correct to six decimal places.
About 530 C.E., Hindu scholar
–
ARYABHATA established
the close approximation
=
for π, and
around 1150 C.E. BH
–
ASKARA, after using this estimate
multiple times in his work, noted that the √
–
10 serves
as a sufficiently accurate approximation for most practical purposes.
In 1429 Arab mathematician JAMSHID AL-KASH
– ı
computed the value of π correct to 16 decimal places,
and in 1610, using a polygon with 2
62 sides to approximate a circle, German mathematician Ludolph van
Ceulen accurately computed the first 35 decimal places
of π. He devoted most of his life to the task.
In 1767 Swiss mathematician JOHANN HEINRICH
LAMBERT proved that π is an IRRATIONAL NUMBER,
thereby establishing that the decimal expansion for π
will never terminate nor fall into a repeating cycle.
Later, in 1873, English scholar WILLIAM SHANKS, after
15 years of work, computed, by hand, the decimal
expansion of π to 607 places. Unfortunately he made
an error in the 527th place, making the decimal expansion thereafter incorrect, but no one noticed the mistake for almost a century.
3,927
–––––
1,250
62,832
––––––
20,000
355
––
113
377
––
120
25
––
8
256
––
81
22
––
7
1
––
7
22
––
7
10
––
71
223
––
71
3
3
2
4
––
√
–
2
C
––
D
kC
––
kD
π =
C
D
392 pi
are also in perspective from a line, and vice versa.
(Although the theorem is stated only for triangles, any
polygonal figure can be subdivided into triangles for
which the theorem applies.)
pi The real number, denoted, π, defined as the ratio
of the circumference C of a CIRCLE of any size drawn in
a plane to its diameter D, is called pi:
Its value is π = 3.141592653589793… That the value of
this ratio is the same for all planar circles is not immediately obvious. To see why this is the case, note that if
any diagram or picture is scaled up or down by a factor
k, then all lengths in that picture change by a factor k. In
particular, for a picture of a circle, the length of its circumference changes to kC and the length of its diameter
to kD. Consequently, the ratio of circumference to
diameter remains unchanged:
=
. That any two
circles in the plane can be thought of as scaled versions
of each other thus explains why the value of π is the
same for all circles. (It is worth mentioning that the
property of scaling used here is a phenomenon of
EUCLIDEAN GEOMETRY and that the value of π is not the
same for all circles in SPHERICAL GEOMETRY or HYPERBOLIC GEOMETRY. For example, the value of π changes
for circles drawn on spheres, ranging in value anywhere
between 2 and 3.141592…)
A study of AREA shows that the area of a circle of
radius r is given by:
A = πr
2
Thus π can also be defined as the ratio of the area of a
circle to its radius squared.
Although it is not standard, one can associate a
value π to other shapes as well. For instance, the ratio of
the perimeter of a square to its width (short diameter) is
the same for all squares. Thus we may say: π square = 4.
(Using the diagonal of the square as the long diameter
produces instead the value π square =
≈ 2.828.) The
ratio of the perimeter of an equilateral triangle to its
diameter (measured as the height of the triangle) is the
same for all equilateral triangles. We have: π triangle =
≈ 3.464. In fact, in precisely this way, by inscribing
polygons with increasingly many sides within a circle
and calculating the associated values of π for these polygons, ARCHIMEDES OF SYRACUSE (ca. 287–212 B.C.E.)
showed that the true value of π for circles lies between
= 3 +
and
= 3 +
. The approximation
, correct to two decimal places, is still often used
today.
Since the time of antiquity, scholars have attempted
to find the exact value of π. The RHIND PAPYRUS from
ancient Egypt (dated ca.1650 B.C.E.) describes a procedure for computing the area of a circle that is
equivalent to using the approximate value
for π.
The Babylonians of the same era used the approximate
value
. Around 150 C.E. Greek astronomer CLAUDIUS
PTOLEMY established that π ≈
, and 300 years later
Chinese scholar ZU CHONGZHI used the improved
approximation π ≈
, correct to six decimal places.
About 530 C.E., Hindu scholar
–
ARYABHATA established
the close approximation
=
for π, and
around 1150 C.E. BH
–
ASKARA, after using this estimate
multiple times in his work, noted that the √
–
10 serves
as a sufficiently accurate approximation for most practical purposes.
In 1429 Arab mathematician JAMSHID AL-KASH
– ı
computed the value of π correct to 16 decimal places,
and in 1610, using a polygon with 2
62 sides to approximate a circle, German mathematician Ludolph van
Ceulen accurately computed the first 35 decimal places
of π. He devoted most of his life to the task.
In 1767 Swiss mathematician JOHANN HEINRICH
LAMBERT proved that π is an IRRATIONAL NUMBER,
thereby establishing that the decimal expansion for π
will never terminate nor fall into a repeating cycle.
Later, in 1873, English scholar WILLIAM SHANKS, after
15 years of work, computed, by hand, the decimal
expansion of π to 607 places. Unfortunately he made
an error in the 527th place, making the decimal expansion thereafter incorrect, but no one noticed the mistake for almost a century.
3,927
–––––
1,250
62,832
––––––
20,000
355
––
113
377
––
120
25
––
8
256
––
81
22
––
7
1
––
7
22
––
7
10
––
71
223
––
71
3
3
2
4
––
√
–
2
C
––
D
kC
––
kD
π =
C
D
392 pi