There is no specific formula for a sector of a circle. There is a formula for its angle (at the centre), its perimeter, its area.
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The area of a sector is the area of the circle multiplied by the fraction of the circle covered by that sector. This is a true statement and correct formula.
If it is a sector of a circle then the arc is the curved part of the circle which forms a boundary of the sector.
That would certainly do it.
It is a sector of the circle
If you're only given the length of the arc, then you can't. You also need to know the fraction of the circle that's in the sector. You can figure that out if you know the angle of the arc, or the radius or diameter of the circle. -- Diameter of the circle = 2 x (radius of the circle) -- Circumference of the circle = (pi) x (Diameter of the circle) -- (length of the arc)/(circumference of the circle) = the fraction of the whole circle that's in the sector or -- (degrees in the arc)/360 = the fraction of the whole circle that's in the sector -- Area of the circle = (pi) x (radius of the circle)2 -- Area of the sector = (Area of the circle) x (fraction of the whole circle that's in the sector)