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# Points to Remember: Areas Related to Circles Notes | EduRev

## Class 10 : Points to Remember: Areas Related to Circles Notes | EduRev

``` Page 1

Circles
and
Areas related
to Circles
Page 2

Circles
and
Areas related
to Circles
Area of
seg AxB (Minor)
=   Area of sector O-AxB
– Area of ? OAB
O
A B
B
A
O
x
Q
Minor
segment
Major
segment
x
O
A B
T = 180
Semicircle
(O-AxB) & (O-AYB)
Y
B
A
O
x
T
Q
A
O
x
B
Minor sector
Major sector
Y
T
O
P
Radius
Distance from the
center of the circle
to the boundary of
the circle.
It is denoted by ‘r’ .
OP is a radius
P
O
Q
Chord
Distance between
any two points on
the circle. A circle
has innitely
many chords.
PQ is a chord
A
B
O
Secant
A line which
intersects the circle
in two distinct
points.
AB is a secant
Segment
Area enclosed by a chord and the arc.
Ex.: segment AxB
O
« «
«
«
«
«
«
«
«
«
«
O
L
Circumference
The length around
the boundary of
the circle
C = 2pr = pd
Tangent
It is a line that
touches the circle
at only one point.
Sector
Area enclosed between two
radii and corresponding
arc of a circle.
B
A
O
Arc
A part of the
boundary or portion
of the circumference.
Length of an arc
× 2pr  =
?
360
× pr
2
?
360
Area of Major
sector
Area of the sector =
Area of the
Major segment
=     pr
2
- Area of the
minor segment
Diameter
A line segment passing through
the center and connecting two
points on the boundary of the
circle. It is denoted by ‘d’ . AB is a
diameter.
Diameter is the longest chord in
the circle. It is double of radius
i.e. d = 2r
=            Area of circle -
Area of the Minor sector
Circles & Areas Related to Circles
Circle is a collection of all points on a plane which are at constant distance (radius) from a ?xed point (center).
CIRCLES & AREAS RELATED TO CIRCLES 25
Area of
a circle
=  pr
2
Page 3

Circles
and
Areas related
to Circles
Area of
seg AxB (Minor)
=   Area of sector O-AxB
– Area of ? OAB
O
A B
B
A
O
x
Q
Minor
segment
Major
segment
x
O
A B
T = 180
Semicircle
(O-AxB) & (O-AYB)
Y
B
A
O
x
T
Q
A
O
x
B
Minor sector
Major sector
Y
T
O
P
Radius
Distance from the
center of the circle
to the boundary of
the circle.
It is denoted by ‘r’ .
OP is a radius
P
O
Q
Chord
Distance between
any two points on
the circle. A circle
has innitely
many chords.
PQ is a chord
A
B
O
Secant
A line which
intersects the circle
in two distinct
points.
AB is a secant
Segment
Area enclosed by a chord and the arc.
Ex.: segment AxB
O
« «
«
«
«
«
«
«
«
«
«
O
L
Circumference
The length around
the boundary of
the circle
C = 2pr = pd
Tangent
It is a line that
touches the circle
at only one point.
Sector
Area enclosed between two
radii and corresponding
arc of a circle.
B
A
O
Arc
A part of the
boundary or portion
of the circumference.
Length of an arc
× 2pr  =
?
360
× pr
2
?
360
Area of Major
sector
Area of the sector =
Area of the
Major segment
=     pr
2
- Area of the
minor segment
Diameter
A line segment passing through
the center and connecting two
points on the boundary of the
circle. It is denoted by ‘d’ . AB is a
diameter.
Diameter is the longest chord in
the circle. It is double of radius
i.e. d = 2r
=            Area of circle -
Area of the Minor sector
Circles & Areas Related to Circles
Circle is a collection of all points on a plane which are at constant distance (radius) from a ?xed point (center).
CIRCLES & AREAS RELATED TO CIRCLES 25
Area of
a circle
=  pr
2
cot ?
tan ?
sin
2
?
+ cos
2
?
= 1 cosec
2
?
- cot
2
?
= 1 sec
2
?
-  tan
2
?
= 1
Proof
Proof
Theorem 2
If two tangents are drawn from an external point
outside the circle; the tangents are equal.
Construction :
Draw a line from point P to the center of the circle
and join OT & OQ also.
To prove : l (PT) = l (PQ)
In ? PQO and ? PTO
OQ = OT
OP = OP
PQO =     PTO = 90
o
Thus , ? PQO       ? PTO
PT = PQ
l (PT) = l (PQ)
?
Common side in both triangles
?
Right-angle Hypotenuse
Side Rule
? (C.S.C.T) Corresponding
Side of Congruent Triangle
?
Radius of same circle
?
Each 90
o
Theorem 1
The tangent at any point of a circle is perpendicular
to the radius through the point of contact.
To prove : OP is perpendicular to xy
i.e OP
-
xy , OQ > OP
If we take any point Q on xy other than P , it will
be outside the circle, because if Q lies inside
the circle, xy will become a secant and
not a tangent, as the tangent touches the circle at
a single point. Likewise , if we take any Point Q,
other than P , it will lie outside the circle.
This proves that OP is the shortest distance
between point O and Tangent xy .
Therefore, OP
-
to tangent xy
T
O
Q
P
=
=
?
?
T
O
Q
P
=
=
?
?
O
P x
?
?
O
P
Q
Y x
?
?
CIRCLES & AREAS RELATED TO CIRCLES 26
Page 4

Circles
and
Areas related
to Circles
Area of
seg AxB (Minor)
=   Area of sector O-AxB
– Area of ? OAB
O
A B
B
A
O
x
Q
Minor
segment
Major
segment
x
O
A B
T = 180
Semicircle
(O-AxB) & (O-AYB)
Y
B
A
O
x
T
Q
A
O
x
B
Minor sector
Major sector
Y
T
O
P
Radius
Distance from the
center of the circle
to the boundary of
the circle.
It is denoted by ‘r’ .
OP is a radius
P
O
Q
Chord
Distance between
any two points on
the circle. A circle
has innitely
many chords.
PQ is a chord
A
B
O
Secant
A line which
intersects the circle
in two distinct
points.
AB is a secant
Segment
Area enclosed by a chord and the arc.
Ex.: segment AxB
O
« «
«
«
«
«
«
«
«
«
«
O
L
Circumference
The length around
the boundary of
the circle
C = 2pr = pd
Tangent
It is a line that
touches the circle
at only one point.
Sector
Area enclosed between two
radii and corresponding
arc of a circle.
B
A
O
Arc
A part of the
boundary or portion
of the circumference.
Length of an arc
× 2pr  =
?
360
× pr
2
?
360
Area of Major
sector
Area of the sector =
Area of the
Major segment
=     pr
2
- Area of the
minor segment
Diameter
A line segment passing through
the center and connecting two
points on the boundary of the
circle. It is denoted by ‘d’ . AB is a
diameter.
Diameter is the longest chord in
the circle. It is double of radius
i.e. d = 2r
=            Area of circle -
Area of the Minor sector
Circles & Areas Related to Circles
Circle is a collection of all points on a plane which are at constant distance (radius) from a ?xed point (center).
CIRCLES & AREAS RELATED TO CIRCLES 25
Area of
a circle
=  pr
2
cot ?
tan ?
sin
2
?
+ cos
2
?
= 1 cosec
2
?
- cot
2
?
= 1 sec
2
?
-  tan
2
?
= 1
Proof
Proof
Theorem 2
If two tangents are drawn from an external point
outside the circle; the tangents are equal.
Construction :
Draw a line from point P to the center of the circle
and join OT & OQ also.
To prove : l (PT) = l (PQ)
In ? PQO and ? PTO
OQ = OT
OP = OP
PQO =     PTO = 90
o
Thus , ? PQO       ? PTO
PT = PQ
l (PT) = l (PQ)
?
Common side in both triangles
?
Right-angle Hypotenuse
Side Rule
? (C.S.C.T) Corresponding
Side of Congruent Triangle
?
Radius of same circle
?
Each 90
o
Theorem 1
The tangent at any point of a circle is perpendicular
to the radius through the point of contact.
To prove : OP is perpendicular to xy
i.e OP
-
xy , OQ > OP
If we take any point Q on xy other than P , it will
be outside the circle, because if Q lies inside
the circle, xy will become a secant and
not a tangent, as the tangent touches the circle at
a single point. Likewise , if we take any Point Q,
other than P , it will lie outside the circle.
This proves that OP is the shortest distance
between point O and Tangent xy .
Therefore, OP
-
to tangent xy
T
O
Q
P
=
=
?
?
T
O
Q
P
=
=
?
?
O
P x
?
?
O
P
Q
Y x
?
?
CIRCLES & AREAS RELATED TO CIRCLES 26
Distance covered by a wheel of a car in one
revolution is equal to the circumference of the wheel.
Total distance covered
Circumference of the wheel
=
?
?
?
?
?
?
When we say  ‘segment’ or ‘sector’ we mean the
‘minor segment’ and the ‘minor sector’ .
Number of revolutions
completed by the wheel
The distances between two parallel
tangents drawn to a circle is equal
to the diameter of the circle
O
P
r
r
Q
If two circles touch internally or externally , the point of
contact lies on the straight line through the two centers.
P
O
r Q
R
{
P O
r
Q
R
Distance between the
centers of two
touching circle. R - r
r + R
PLEASE KEEP IN MIND
Areas related to a Circle - Part 1
Amazing tricks to remember
circle formula
Introduction to circles
Tangent to a circle
Circles- Theorem - part 1
SCcan the QR Codes to watch our free videos
To calculate the shaded/ required area in any gure which is
related to circle, rst observe all the gures which are
involved in the diagram like square, rectangle, triangle, and
circle. And then analyse accordingly, the areas which are to
be added or subtracted to get required shaded region.
CIRCLES & AREAS RELATED TO CIRCLES 27
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