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

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 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 180°, then multiply the quotient by π times the radius squared.

1 answer

To derive the formula for the area of a sector, we can start with the understanding that the area of a whole circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius.

  1. A full circle corresponds to an angle of \( 360^\circ \).
  2. The area of a sector of that circle is a fraction of the total area based on its angle \( \theta \).

The fraction of the angle of the sector to the full angle of the circle is \( \frac{\theta}{360^\circ} \).

Thus, to find the area of the sector \( A \), we multiply the total area of the circle by that fraction:

\[ A = \frac{\theta}{360^\circ} \times \pi r^2 \]

So the correct response is:

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