1. CIRCLES
PRESENTED BY
ADAMYA SHYAM

2. CIRCLE
DEFINITION
A Circle is a simple shape of Euclidean geometry that is the set
of points in the plane that are equidistant from a given
point, the centre. The distance between any of the points on the
circle and the centre is called the radius. A circle is a simple
closed curve which divides the plane into 3 regions:
Interior, Exterior and On The Circle . In everyday use, the term
"circle" may be used interchangeably to refer to either the
boundary of the figure, or to the whole figure including its
interior; in strict technical usage, the circle is the former and the
latter is called a disk. A circle can be defined as the curve traced
out by a point that moves so that its distance from a given point
is constant. A circle may also be defined as a special ellipse in
which the two foci are coincident and the eccentricity is 0.
Circles are conic sections attained when a right circular cone is
intersected by a plane perpendicular to the axis of the cone.

3. CIRCLE
HISTORY
The word "circle" derives from the Greek, kirkos "a circle," from the
base Ker- which means to turn or bend. The origins of the words
"circus" and "circuit" are closely related. The circle has been known
since before the beginning of recorded history. Natural circles
would have been observed, such as the Moon, Sun, and a short
plant stalk blowing in the wind on sand, which forms a circle shape
in the sand. The circle is the basis for the wheel, which, with related
inventions such as gears, makes much of modern civilization
possible. In mathematics, the study of the circle has helped inspire
the development of geometry, astronomy, and calculus. Early
science, particularly geometry and astrology and astronomy was
connected to the divine for most medieval scholars, and many
believed that there was something intrinsically "divine" or "perfect"
that could be found in circles.

4. CIRCLE
TERMINOLOGY
Chord: A line segment whose endpoints lie on the circle.
Diameter: A line segment whose endpoints lie on the Circle
and which passes through the centre.
Radius: Half of Diameter
Arc: Any connected part of the circle's circumference.
Sector: A region bounded by two radii and an arc lying
between the radii.
Segment: A region bounded by a chord and an arc lying
between the chord's endpoints.

5. CIRCLE
CHORD
Chords are equidistant from the centre of a circle if and only if
they are equal in length. If a central angle and an inscribed angle
of a circle are subtended by the same chord and on the same side
of the chord, then the central angle is twice the inscribed angle. If
two angles are inscribed on the same chord and on the same side
of the chord, then they are equal. If two angles are inscribed on
the same chord and on opposite sides of the chord, then they are
supplemental. An inscribed angle subtended by a diameter is a
right angle. The diameter is the longest chord of the circle. If the
intersection of any two perpendicular chords divides one chord
into lengths a and b and divides the other chord into lengths c
and d, then a2 + b2 + c2 + d2 equals the square of the diameter.
The distance from a point on the circle to a given chord times the
diameter of the circle equals the product of the distances from
the point to the ends of the chord.

6. CIRCLE
DIAMETER
In geometry, the diameter of a circle is any straight line
segment that passes through the center of the circle and
whose endpoints are on the boundary of the circle. The
diameters are the longest chords of the circle. In this sense
one speaks of diameter rather than a diameter, because all
diameters of a circle have the same length, this being twice
the radius. For a convex shape in the plane, the diameter is
defined to be the largest distance that can be formed
between two opposite parallel lines tangent to its
boundary, and the width is defined to be the smallest such
distance. For a curve of constant width such as the Reuleaux
triangle, the width and diameter are the same because all
such pairs of parallel tangent lines have the same distance.
See also Tangent lines to circles.

7. CIRCLE
RADIUS
In classical geometry, a radius of a circle is any line segment
from its center to its perimeter. By extension, the radius of a
circle or sphere is the length of any such segment, which is half
the diameter. If the object does not have an obvious center, the
term may refer to its circum radius, the radius of its
circumscribed circle . In either case, the radius may be more
than half the diameter, which is usually defined as the
maximum distance between any two points of the figure. The in
radius of a geometric figure is usually the radius of the largest
circle or sphere contained in it. The inner radius of a ring, tube
or other hollow object is the radius of its cavity. For regular
polygons, the radius is the same as its circumradius.The name
comes from Latin radius, meaning "ray" but also the spoke of a
chariot wheel.

8. CIRCLE
ARC & SECTOR
In geometry, an arc is a closed segment of a differentiable curve
in the two-dimensional plane; for example, a circular arc is a
segment of the circumference of a circle. If the arc is part of a
great circle , it is called a great arc.
A circular sector or circle sector, is the portion of a disk
enclosed by two radii and an arc, where the smaller area is
known as the minor sector and the larger being the major
sector. In the diagram, θ is the central angle in radians, the
radius of the circle, and is the arc length of the minor sector. A
sector with the central angle of 180° is called a semicircle.
Sectors with other central angles are sometimes given special
names, these include quadrants (90°), sextants (60°) and
octants (45°).

9. CIRCLE
SEGMENT
In geometry, a circular segment is an area of a circle informally
defined as an area which is "cut off" from the rest of the circle
by a chord. The circle segment constitutes the part between
the secant and an arc, excluding of the circle's center.
SEGMENT
CHORD
RADIUS
CENTRE
DIAMETER
SECTOR

10. CIRCLE
THEOREMS
The chord theorem states that if two chords, CD and EB, intersect
at A, then CA × DA = EA × BA. If a tangent from an external point D
meets the circle at C and a secant from the external point D meets
the circle at G and E respectively, then DC2 = DG × DE. If two
secants, DG and DE, also cut the circle at H and F respectively, then
DH × DG = DF × DE. The angle between a tangent and chord is
equal to one half the subtended angle on the opposite side of the
chord. If the angle subtended by the chord at the centre is 90
degrees then l = r√2, where l is the length of the chord and r is the
radius of the circle. If two secants are inscribed in the circle as
shown at right, then the measurement of angle A is equal to one
half the difference of the measurements of the enclosed arcs.