Points to Remember: Areas Related to Circles Notes | EduRev

Mathematics Chart book, A Quick guide

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 innitely  
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 innitely  
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 innitely  
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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