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How do you derive the formula for the area of a sector?(1 point)

Responses

Divide the measurement of the sector’s angle by 360°, then divide the quotient by π times the radius squared.
Divide the measurement of the sector’s angle by 360 degrees , then divide the quotient by pi times the radius squared.

Divide the measurement of the sector’s angle by 360°, then multiply the quotient by π times the radius squared.
Divide the measurement of the sector’s angle by 360 degrees , then multiply the quotient by pi times the radius squared.

Divide the measurement of the sector’s angle by 180°, then multiply the quotient by π times the radius squared.
Divide the measurement of the sector’s angle by 180 degrees , then multiply the quotient by pi times the radius squared.

Divide the measurement of the sector’s angle by 360°, then multiply the quotient by π times the radius.

1 answer

To derive the formula for the area of a sector of a circle, you can use the following approach:

  1. 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.

  2. Proportion of the sector: A sector is a fraction of the full circle based on its angle. The angle of the sector in degrees can be represented as \( \theta \). The ratio of the sector's angle to the full angle of a circle (360°) is \( \frac{\theta}{360°} \).

  3. Calculating the area of the sector: To find the area of the sector, you take the area of the full circle and multiply it by the fraction that represents the sector: \[ \text{Area of sector} = \frac{\theta}{360°} \times \text{Area of full circle} \] This gives: \[ \text{Area of sector} = \frac{\theta}{360°} \times \pi r^2 \]

Thus, the correct way to derive the formula for the area of a sector is: Divide the measurement of the sector’s angle by 360°, then multiply the quotient by π times the radius squared.