Chapter 12

Area Related to Circles

Circle: A circle is a collection of points in a plane which are the same distance from a given point called the center. The fixed distance from the centre is called radius of the circle.

Diameter of Circle: A chord which passes through the centre of the circle is called the diameter of the circle.

Diameter = 2 x Radius

The common distance of the points of a circle from its center is called its radius.

In the figure, O is the centre. OA, OB and OC are the radii of the circle and AB is the diameter.

Circumference of the Circle: The perimeter of a circle is known as its circumference of a circle. Let r be the radius of the circle.

Circumference of circle =

Area of Circle: Let r be the radius of the circle, then

Area of the circle =

Area between two concentric Circles: Let R and r be the radii of two concentric circles (R > r).

Then the area between the circles = Area of Outer Circle – Area of smaller circle

A =

Some Results:

(i) If two circles touch externally, then the distance between the centers of the circles is the sum of their radii.

(ii) If two circles touch internally, then the distance between the centers of the circles is the difference of their radii.

(iii) Distance covered by a wheel (circle) is equal to the circumference of the wheel.

Sector of a Circle: The area enclosed by the arc and two radii of a circle is called the sector of the circle.

Here, the radii of the circle divides the whole circle in two regions, these regions are called sectors.

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

In the figure, OAXBO is the minor sector.

Also, if, then sector of circle will be minor

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

In the figure, OAYBO is the minor sector.

Also, if, then sector of circle will be minor

Arc length and Area of Sector of a Circle:

(1) Arc length

(2) Area of sector

Remarks:

(1) Angle described by minute hand in 60 minutes = 360°

Angle described by minute hand in 1 minutes = 360°/60 = 6°

(2) Angle described by the hour hand in one hour = 360°/12 = 30°

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

Minor Segment: If the boundary of segment is minor arc of a circle, then the corresponding segment is called a minor segment.

Major Segment: If the boundary of segment is major arc of a circle, then the corresponding segment is called a major segment.

In the figure, region ABXA is minor segment and ABYA is major segment.

Area of Segment of Circle:

(1) Area of minor segment ABXA = Area of sector OAXBO – Area of triangle OAB

Area of minor segment ABXA =

(2) Area of major segment = Area of circle – Area of minor segment