To derive the formula for the area of a sector, we can start with the following steps:
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Understand the full circle area: The area \( A \) 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 represented by the sector: A sector of a circle is essentially a portion of the full circle, defined by an angle \( \theta \) (in degrees). The fraction of the circle that the sector represents is: \[ \text{Fraction} = \frac{\theta}{360^\circ} \]
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Calculate the area of the sector: To find the area \( A_{sector} \) of the sector, multiply the area of the full circle by the fraction that the sector represents: \[ A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]
Thus, the correct derivation gives the area of a sector as: \[ A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]
Given your response options, the correct choice is:
- Divide the measurement of the sector’s angle by 360°, then multiply the quotient by π times the radius squared.
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