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

Question

Responses

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 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 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 divide the quotient by π times the radius squared.

Answer 1

To derive the formula for the area of a sector of a circle, you start from the concept that the area of a full circle is given by the formula:

$$A=\pi r^2$$

where $r$ is the radius of the circle.

A sector is a portion of the circle defined by a central angle $\theta$. The angle $\theta$ is typically measured in degrees. To find the area of the sector, you need to determine what fraction of the full circle this angle represents.

1. The full circle corresponds to ${360}^{\circ }$.

2. The fraction of the circle that the sector represents is given by $\frac{\theta }{{360}^{\circ }}$.

3. To find the area of the sector, multiply this fraction by the total area of the circle:

$$\text{Area of sector}=\left(\frac{\theta }{{360}^{\circ }}\right)\times \pi r^2$$

Thus, the correct response 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.