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The area of a sector of a circle is calculated by multiplying the circle's area by the ratio of the measure of the central angle to 360^(∘).
The area of the sector is calculated by the following formula.
Area of Sector = θ/360^(∘) * π r^2
From the fact that 2πrad equals 360^(∘), an equivalent formula can be written if the central angle is given in radians.
Area of Sector &= θ/2π * π r^2 &⇓ Area of Sector &= θ/2* r^2
Since the measure of an arc is equal to the measure of its central angle, the arc AB measures θ. Therefore, by substituting mAB for θ, another version of the formula is obtained which can also be written in degrees or radians.
Area of Sector = mAB/360^(∘) * π r^2
or
Area of Sector = mAB/2* r^2
Since a circle measures 360^(∘), this sector represents θ360^(∘) of ⊙ C. Therefore, the ratio of the area of a sector to the area of the whole circle is proportional to θ360^(∘). Area of Sector/Area of Circle = θ/360^(∘)
Recall that the area of a circle is π r^2. By substituting it into the equation and solving for the area of a sector, the desired formula can be obtained.
Area of Sector/π r^2 = θ/360^(∘)
Therefore, the area of a sector of a circle can be found by using the following formula.
Area of Sector = θ/360^(∘) * π r^2