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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
Distance from the
center of the circle
to the boundary of
the circle.
It is denoted by ‘r’ .
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
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
Distance from the
center of the circle
to the boundary of
the circle.
It is denoted by ‘r’ .
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
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
?
?
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
Distance from the
center of the circle
to the boundary of
the circle.
It is denoted by ‘r’ .
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
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
?
?
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
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
CIRCLES & AREAS RELATED TO CIRCLES 27
```
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