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Introduction to Circles
A circle is a simple shape of Euclidean geometry consisting of
those points in a plane which are the
same distance from a given point called the centre. The common
distance of the points of a circle
from its center is called its radius. A diameter is a line
segment whose lie on the circle and whichpasses through the centre
of the circle. The length of a diameter is twice the length of the
radius. A
circle is never a because it has no sides or vertices
Circles are simple closed curves which divide the plane into two
regions, an interior and an exterior.
In everyday use the term "circle" may be used interchangeably to
refer to either the boundary of the
figure (known as the perimeter) or to the whole figure including
its interior, but in strict technical
usage "circle" refers to the perimeter while the interior of the
circle is called a disk. The
circumference of a circle is the perimeter of the circle
(especially when referring to its length).
A circle is a special ellipse in which the two foci are
coincident. Circles are conic sections attained
when a right circular cone is intersected with a plane
perpendicular to the axis of the cone.
A chord of a circle is a line segment whose two endpoints lie on
the circle. The diameter, passing
through the circle's centre, is the largest chord in a circle. A
tangent to a circle is a straight line thattouches the circle at a
single point. A secant is an extended chord: a straight line
cutting the circle at
two points.
An arc of a circle is any connected part of the circle's
circumference. A sector is a region bounded
by two radii and an arc lying between the radii, and a segment
is a region bounded by a chord and
an arc lying between the chord's endpoints.
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(b) Pi oris a mathematical constant related to circles. Define
and write a brief history of
A Brief History of Pi ()
Pi or is a mathematical constant whose value is the ratio of any
circle's circumference to its
diameter in Euclidean space; this is the same value as the ratio
of a circle's area to the square of its
radius. The ratio of the circumference to the diameter of a
circle is constant (namely, pi) has beenrecognized for as long as
we have written records. A ratio of 3:1 appears in the following
biblical
verse:
And he made a molten sea, ten cubits from the one brim to the
other: it was round all about, and his
height was five cubits: and a line of thirty cubits did compass
it about. (I Kings 7, 23; II Chronicles
4, 2.)
It is approximately equal to 3.14159 in the usual decimal
notation (see the table for its
representation in some other bases). is one of the most
important mathematical and physical
constants: many formulae from mathematics, science, and
engineering involve .
is an irrational number, which means that its value cannot be
expressed exactly as a fractionm/n,
where m and n are integers. Consequently, its decimal
representation never ends or repeats. It is also
a transcendental number, which means that no finite sequence of
algebraic operations on integers(powers, roots, sums, etc.) can be
equal to its value; proving this was a late achievement in
mathematical history and a significant result of 19th century
German mathematics. Throughout the
history of mathematics, there has been much effort to determine
more accurately and to
understand its nature; fascination with the number has even
carried over into non-mathematical
culture.
The Greek letter , often spelled outpi in text, was adopted for
the number from the Greek word for
perimeter"", first by William Jones in 1707, and popularized by
Leonhard Eulerin1737. The constant is occasionally also referred to
as the circular constant, Archimedes' constant or
Ludolph's number.
The ancient Babylonians generally calculated the area of a
circle by taking 3 times the square of its
radius ( =3), but one Old Babylonian tablet (from ca. 1900-1680
BCE) indicates a value of 3.125
for pi. Ancient Egyptians calculated the area of a circle by the
following formula (where dis thediameter of the circle):
This yields an approximate value of 3.1605 for pi.
The first theoretical calculation of a value of pi was that of
Archimedes of Syracuse (287-212 BCE),
one of the most brilliant mathematicians of the ancient world.
Archimedes worked out that 223/71
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