Area of Sector of a Circle
- The sector of a circle is the portion of the circle which is formed by the enclosing two arms which are the radii of circle and an arc of the circle. And the area enclosed by that sector is called as the area of the sector of the circle.
- In following figure, the portion of circle AOBMA indicates the sector of circle which is called as the minor sector of circle.
- And the remaining portion of the circle AOBNA represents the major sector of the circle.
- We know that, if r is the radius of circle, then the area of circle will be,
Area of circle = π*r2
- And the area of the semicircle of the same circle will be,
Area of semicircle = (π*r2)/ 2
- And the area of the quarter part of the same circle will be,
Area of quarter part of circle = (π*r2)/ 4
- The following figure shows the area enclosed by whole circle, semicircle and quarter part of the circle.
- As the total circle include total angle at the center of circle which is 3600.
- Also, the semicircle includes angle 1800 at the center of the circle.
- And the quarter part of circle includes angle 900 at the center of circle.
- From the following figure we can rewrite the formulae of areas of circle, semicircle and quarter part of circle also.
Thus,
- Area of circle = π*r2
= (3600/3600) * π*r2
- Area of semicircle = (π*r2)/ 2
= (1800/3600) * π*r2
- Area of quarter part of circle = (π*r2)/ 4
= (900/3600) * π*r2
Thus, we can generalize the above formula of area as:
Area of any portion of circle = (ϴ/3600) * π*r2
To find the area of sector of a circle:
- In figure below the portion AOBMA represents the area of minor sector of circle.
- And ϴ be the angle between the two radii which forms the minor sector of circle as shown in figure. Here, as the value of ϴ increases the area of sector also increases.
- The sector having large angle ϴ is called as major sector and the sector having small angle ϴ as compared to major sector then it is called as minor sector of the circle.
- Above we have written all the formulae of areas in terms of angle.
- In similar way, in terms of ϴ we can write the area of minor sector and major sector also.
Thus,
Area of minor sector AOBMA of circle = (angle enclosed by sector/ 3600) * π*r2
= (ϴ/ 3600) * π*r2
Thus, area of sector of circle when angle ϴ is in degree,
Area of sector of circle =(ϴ/ 3600) * π*r2
And the area of sector of a circle when angle is given in radian,
Area of sector of circle = ½*r2*ϴ
Hence proved.
