168 MATHEMATICS
CHAPTER 10
CIRCLES
10.1 Introduction
You may have come across many objects in daily life, which are round in shape, such
as wheels of a vehicle, bangles, dials of many clocks, coins of denominations 50 p,
Re 1 and Rs 5, key rings, buttons of shirts, etc. (see Fig.10.1). In a clock, you might
have observed that the second’s hand goes round the dial of the clock rapidly and its
tip moves in a round path. This path traced by the tip of the second’s hand is called a
circle. In this chapter, you will study about circles, other related terms and some
properties of a circle.
Fig. 10.1
2020-21
,CIRCLES 169
10.2 Circles and Its Related Terms: A Review
Take a compass and fix a pencil in it. Put its pointed
leg on a point on a sheet of a paper. Open the other
leg to some distance. Keeping the pointed leg on the
same point, rotate the other leg through one revolution.
What is the closed figure traced by the pencil on
paper? As you know, it is a circle (see Fig.10.2). How
did you get a circle? You kept one point fixed (A in
Fig.10.2) and drew all the points that were at a fixed
distance from A. This gives us the following definition:
The collection of all the points in a plane,
Fig. 10.2
which are at a fixed distance from a fixed point in
the plane, is called a circle.
The fixed point is called the centre of the circle
and the fixed distance is called the radius of the
circle. In Fig.10.3, O is the centre and the length OP
is the radius of the circle.
Remark : Note that the line segment joining the
centre and any point on the circle is also called a
radius of the circle. That is, ‘radius’ is used in two Fig. 10.3
senses-in the sense of a line segment and also in the
sense of its length.
You are already familiar with some of the
following concepts from Class VI. We are just
recalling them.
A circle divides the plane on which it lies into
three parts. They are: (i) inside the circle, which is
also called the interior of the circle; (ii) the circle
and (iii) outside the circle, which is also called the Fig. 10.4
exterior of the circle (see Fig.10.4). The circle and
its interior make up the circular region.
If you take two points P and Q on a circle, then the line segment PQ is called a
chord of the circle (see Fig. 10.5). The chord, which passes through the centre of the
circle, is called a diameter of the circle. As in the case of radius, the word ‘diameter’
is also used in two senses, that is, as a line segment and also as its length. Do you find
any other chord of the circle longer than a diameter? No, you see that a diameter is
the longest chord and all diameters have the same length, which is equal to two
2020-21
, 170 MATHEMATICS
times the radius. In Fig.10.5, AOB is a diameter of
the circle. How many diameters does a circle have?
Draw a circle and see how many diameters you can
find.
A piece of a circle between two points is called
an arc. Look at the pieces of the circle between two
points P and Q in Fig.10.6. You find that there are Fig. 10.5
two pieces, one longer and the other smaller
(see Fig.10.7). The longer one is called the major
arc PQ and the shorter one is called the minor arc
PQ. The minor arc PQ is also denoted by PQ and
, where R is some point on
the major arc PQ by PRQ
the arc between P and Q. Unless otherwise stated,
stands for minor arc PQ. When P and
arc PQ or PQ
Q are ends of a diameter, then both arcs are equal Fig. 10.6
and each is called a semicircle.
The length of the complete circle is called its
circumference. The region between a chord and
either of its arcs is called a segment of the circular
region or simply a segment of the circle. You will find
that there are two types of segments also, which are
the major segment and the minor segment
(see Fig. 10.8). The region between an arc and the
two radii, joining the centre to the end points of the
arc is called a sector. Like segments, you find that Fig. 10.7
the minor arc corresponds to the minor sector and the major arc corresponds to the
major sector. In Fig. 10.9, the region OPQ is the minor sector and remaining part of
the circular region is the major sector. When two arcs are equal, that is, each is a
semicircle, then both segments and both sectors become the same and each is known
as a semicircular region.
Fig. 10.8 Fig. 10.9
2020-21
CHAPTER 10
CIRCLES
10.1 Introduction
You may have come across many objects in daily life, which are round in shape, such
as wheels of a vehicle, bangles, dials of many clocks, coins of denominations 50 p,
Re 1 and Rs 5, key rings, buttons of shirts, etc. (see Fig.10.1). In a clock, you might
have observed that the second’s hand goes round the dial of the clock rapidly and its
tip moves in a round path. This path traced by the tip of the second’s hand is called a
circle. In this chapter, you will study about circles, other related terms and some
properties of a circle.
Fig. 10.1
2020-21
,CIRCLES 169
10.2 Circles and Its Related Terms: A Review
Take a compass and fix a pencil in it. Put its pointed
leg on a point on a sheet of a paper. Open the other
leg to some distance. Keeping the pointed leg on the
same point, rotate the other leg through one revolution.
What is the closed figure traced by the pencil on
paper? As you know, it is a circle (see Fig.10.2). How
did you get a circle? You kept one point fixed (A in
Fig.10.2) and drew all the points that were at a fixed
distance from A. This gives us the following definition:
The collection of all the points in a plane,
Fig. 10.2
which are at a fixed distance from a fixed point in
the plane, is called a circle.
The fixed point is called the centre of the circle
and the fixed distance is called the radius of the
circle. In Fig.10.3, O is the centre and the length OP
is the radius of the circle.
Remark : Note that the line segment joining the
centre and any point on the circle is also called a
radius of the circle. That is, ‘radius’ is used in two Fig. 10.3
senses-in the sense of a line segment and also in the
sense of its length.
You are already familiar with some of the
following concepts from Class VI. We are just
recalling them.
A circle divides the plane on which it lies into
three parts. They are: (i) inside the circle, which is
also called the interior of the circle; (ii) the circle
and (iii) outside the circle, which is also called the Fig. 10.4
exterior of the circle (see Fig.10.4). The circle and
its interior make up the circular region.
If you take two points P and Q on a circle, then the line segment PQ is called a
chord of the circle (see Fig. 10.5). The chord, which passes through the centre of the
circle, is called a diameter of the circle. As in the case of radius, the word ‘diameter’
is also used in two senses, that is, as a line segment and also as its length. Do you find
any other chord of the circle longer than a diameter? No, you see that a diameter is
the longest chord and all diameters have the same length, which is equal to two
2020-21
, 170 MATHEMATICS
times the radius. In Fig.10.5, AOB is a diameter of
the circle. How many diameters does a circle have?
Draw a circle and see how many diameters you can
find.
A piece of a circle between two points is called
an arc. Look at the pieces of the circle between two
points P and Q in Fig.10.6. You find that there are Fig. 10.5
two pieces, one longer and the other smaller
(see Fig.10.7). The longer one is called the major
arc PQ and the shorter one is called the minor arc
PQ. The minor arc PQ is also denoted by PQ and
, where R is some point on
the major arc PQ by PRQ
the arc between P and Q. Unless otherwise stated,
stands for minor arc PQ. When P and
arc PQ or PQ
Q are ends of a diameter, then both arcs are equal Fig. 10.6
and each is called a semicircle.
The length of the complete circle is called its
circumference. The region between a chord and
either of its arcs is called a segment of the circular
region or simply a segment of the circle. You will find
that there are two types of segments also, which are
the major segment and the minor segment
(see Fig. 10.8). The region between an arc and the
two radii, joining the centre to the end points of the
arc is called a sector. Like segments, you find that Fig. 10.7
the minor arc corresponds to the minor sector and the major arc corresponds to the
major sector. In Fig. 10.9, the region OPQ is the minor sector and remaining part of
the circular region is the major sector. When two arcs are equal, that is, each is a
semicircle, then both segments and both sectors become the same and each is known
as a semicircular region.
Fig. 10.8 Fig. 10.9
2020-21
