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Circles (Area and Perimeter)

Important formula of Circles:

1 . Area of circle = π r2

2 . Circumference = 2πr

3 . Diameter = 2 x radius

4 . Area of circle = π
d2/4
, where d is Diameter .

5 . Area of Semicircle = π
r2/2


6 . Area of a Quadrant of circle = π
r2 /4


7 . Area in closed by two concentric circle = π R2 - π r2

= π( R2 - r2) = π(R - r)(R + r)




8 . (a) . If two circle touch internally then the distance between their centers is equal to the difference of their radii .

(b) . If two circles touch externally then the distance between their centers is equal to the sum of their radii .

(c) . Distance travelled by a rotating wheel in one revoluation is equal to the circumference of the wheel .



Sector of a Circle and its Area

In general their are two sector in a circle minor and major .

Minor sector : A sector of a circle is called a minor sector if the minor arc of the circle is a part of its boundary .

Major sector : A sector of a circle is called a major sector if the major arc of the circle is a part of its boundary.

Important Points :

(a) . The sum of arcs of major and minor sectors of a circle is equal to the circumference of the circle .

(b) . The sum of the areas of major and minor sector of a circle is equal to the area of the circle .


Important formulas of sector of a circle

1 . Area of sector =
θ/360°
x π r2

2 . Area of sector =
1/2
x lr, where l is length of arc

3 . Area of sector =
1/2
x r2θ


4 . Length of arc (l) =
θ/ 360°
x 2πr

Segment of a circle

The region enclosed by an arc and a chord is called the segment of the circle .

Their are two segments in a circle major segment and minor segment .

Major segment : A segment corresponding a major arc of a circle is called the major segment .

Minor segment : A segment corresponding a minor arc of a circle is called the minor segment .



Formula of segment of a circle



AOBA is minor segment , APB is a triangle and PAOB is a sector ,

Area of segment AOBA = Area of sector AOBP - Area of triangle APB

=
θπr2/360°
- r2 sin
θ/2
cos
θ/2


= [
θπ/360°
- sin
θ/2
cos
θ/2
]r2