To derive the formula for the area of a sector, you can follow these steps:
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Understanding the full circle: The area of a full circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle.
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Fraction of the circle: A sector of a circle (which is essentially a "slice" of the circle) represents a fraction of the full circle based on its angle \( \theta \) (in degrees). The fraction of the full circle that the sector occupies is given by \( \frac{\theta}{360^\circ} \).
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Area of the sector: To find the area of the sector, you multiply the area of the entire circle by the fraction of the circle that the sector occupies: \[ \text{Area of sector} = \left(\frac{\theta}{360^\circ}\right) \times \text{Area of full circle} \] \[ \text{Area of sector} = \left(\frac{\theta}{360^\circ}\right) \times \pi r^2 \]
Based on this reasoning, the correct choice among the options provided is:
Divide the measurement of the sector’s angle by 360°, then multiply the quotient by π times the radius squared.