What is the formula for the area of a sector?

The formula for the area of a sector is derived based on the proportion of the angle of the sector to the full angle of a circle, which is 360 degrees. The total area of a circle is given by the formula (A = \pi r^2).

To find the area of a sector, you take the proportion of the angle in degrees for the sector, (x°), divided by the total degrees in a circle, which is 360. Thus, the fraction of the area of the circle that represents the sector is ( \frac{x°}{360} ).

To find the actual area of the sector, you multiply this fraction by the total area of the circle, giving the complete formula:

[ A = \frac{x°}{360} \times (\pi r^2) ]

The correct answer captures this relationship perfectly. The other choices misrepresent how the area of a sector is calculated, either altering the basic structure or improperly relating the angle to the area formula. This clarity in understanding how to derive the sector's area from the known properties of circles is critical in effectively applying these formulas in practical problems.