392 pi

4

––

√

–

2

C

––

D

kC

––

kD

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 Cof 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 immedi-

ately 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 cir-

cumference 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 πis the

same for all circles. (It is worth mentioning that the

property of scaling used here is a phenomenon of

E

UCLIDEAN GEOMETRY

and that the value of πis not the

same for all circles in

SPHERICAL GEOMETRY

or

HYPER

-

BOLIC GEOMETRY

. For example, the value of πchanges

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 ris given by:

A= πr2

Thus πcan 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 πto 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: πsquare = 4.

(Using the diagonal of the square as the long diameter

produces instead the value πsquare = ≈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: πtriangle =

≈3.464. In fact, in precisely this way, by inscribing

polygons with increasingly many sides within a circle

and calculating the associated values of πfor these poly-

gons, A

RCHIMEDES OF

S

YRACUSE

(ca. 287–212

B

.

C

.

E

.)

showed that the true value of πfor 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 R

HIND PAPYRUS

from

ancient Egypt (dated ca.1650

B

.

C

.

E

.) describes a pro-

cedure 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 C

LAUDIUS

P

TOLEMY

established that

π

≈, and 300 years later

Chinese scholar Z

U

C

HONGZHI

used the improved

approximation

π

≈, correct to six decimal places.

About 530

C

.

E

., Hindu scholar –

A

RYABHATA

established

the close approximation = for

π

, and

around 1150

C

.

E

.B

H

–

ASKARA

, after using this estimate

multiple times in his work, noted that the √

–

10 serves

as a sufficiently accurate approximation for most prac-

tical purposes.

In 1429 Arab mathematician J

AMSHID AL

-K

ASH

–

ı

computed the value of πcorrect to 16 decimal places,

and in 1610, using a polygon with 262 sides to approxi-

mate 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 J

OHANN

H

EINRICH

L

AMBERT

proved that πis 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 W

ILLIAM

S

HANKS

, after

15 years of work, computed, by hand, the decimal

expansion of πto 607 places. Unfortunately he made

an error in the 527th place, making the decimal expan-

sion thereafter incorrect, but no one noticed the mis-

take 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