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Areas of Sector and Segment of a Circle

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Areas of Sector and Segment of a Circle - Lesson Summary

Circle
Locus of a point which moves in a plane in such a way that its distance from a fixed point always constant. The fixed point is called the centre and constant distance is called radius of the circle.

If r be the radius of the circle then,
1) Circumference = 2πr or πd, where d = 2r is the diameter of the circle.
2) Area = πr 2 or π d 2 4 .
3) Area of a semi-circle = π r 2 2
4) Area of a quadrant of a circle = π r 2 4

The arc of a circle is a part of the circumference of the circle.

Area enclosed between two concentric circles
If R and r are two radii of concentric circles, then Area enclosed between two concentric circles = πR 2 - πr 2 = π(R 2 - r 2) = π(R + r)(R - r)
  1. If two circles touch internally, then the difference of their radii is equal to the distance between their centres.
  2. If two circles touch externally, then the sum of their radii is equal to the distance between their centres.
  3. Distance moved by a rotating wheel in one revolution is equal to the Circumference of the circle.
  4. The number of revolutions completed by a rotating wheel in one minute = Distance moved in one minute Circumference

Sector of a circle:
The circular region enclosed by two radii and the corresponding arc of a circle is called the sector of a circle.

Here OAPB is called minor sector and OAQB is called the major sector.

1) Sum of the arcs of major and minor sectors of a circle is equal to circumference of the circle.
2) Sum of the areas of major and minor sectors of a circle is equal to area of the circle.

Angle of the sector:
The angle subtended by the corresponding arc of the sector at the centre of the circle is called the angle of the sector.

Area of a Sector of angle θ =  θ 360 x πr 2 or 1 2 × length of arc × radius = 1 2 lr

Length of an arc of angle θ =  θ 360 x 2πr.

Segment of a circle:
The circular region enclosed between a chord and the corresponding arc is called the segment of a circle.

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

Major segment
A segment corresponding a major arc of a circle is called as major segment.


Here APB is called minor segment and AQB is called major segment.

The area of a segment is the area of the corresponding sector minus the area of the corresponding triangle.

Area of Segment APB = Area of Sector OAPB – Area of ΔOAB = θ 360 x πr 2 – 1 2 r 2 sin θ

Angle described by minute hand in 60 minutes = 360°.

Angle described by minute hand in one minute = ( 360 60 )° = 6°.

Angle described by hour hand in 12 hours = 360°.

Angle described by hour hand in one hour = ( 360 12 )° = 30°.

Angle described by hour hand in one minute = ( 30 60 )° = 1 2 °.

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