Areas Related to Circles
To understand the concepts of length of arc and the area of the sector of the circle, look at the following video.
Some important formulae to remember are:
Area of a sector of angleθ |
In case of quadrant, θ = 90°. Therefore,
Area of a quadrant |
In case of semicircle, θ = 180°. Therefore,
Area of a semicircle |
Length of an arc of a sector of angle θ |
Perimeter of a Sector = $l+2r$ |
Relation between the area of sector and the length of an arc:
Look at the following figure.
Here, O is the centre of the circle of radius r and APB is an arc of length l.
Clearly, measure of arc APB = Central angle = θ
Let the area of sector OAPB be A.
Now, we have
From length of arc and area of corresponding sector, we can also conclude that.
These formulae are very helpful to calculate various attributes related to circle.
Let us solve some examples to learn the concept better.
Example 1:
Find the area of the sectors OACB and OADB. Also find the length of the arc ACB. (Take π =)
Solution:
Given, radius of the circle, r = 6.3 cm
Angle of sector OACB, θ = 40°
∴ Area of sector OACB = $\frac{\theta}{360}\times {\mathrm{\pi r}}^{2}$
= 13.86...
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