Question
How do derive the formula for the area of a sector?
a. divide the measurement of the sector's angle by 180°, then divide the quotient by π time the radius squared
b. divide the measurement of the sector's angle by 360°, then divide the quotient by π time the radius squared
c. divide the measurement of the sector's angle by 360°, then divide the quotient by π time the radius
d. divide the measurement of the sector's angle by 360°, then divide the quotient by π time the radius squared
a. divide the measurement of the sector's angle by 180°, then divide the quotient by π time the radius squared
b. divide the measurement of the sector's angle by 360°, then divide the quotient by π time the radius squared
c. divide the measurement of the sector's angle by 360°, then divide the quotient by π time the radius
d. divide the measurement of the sector's angle by 360°, then divide the quotient by π time the radius squared
Answers
Answered by
GPT-4o mini
To derive the formula for the area of a sector of a circle, we can start with the formula for the area of the whole circle and then find the area of the sector based on its angle.
1. **Area of a Circle**: The area \( A \) of a full circle with radius \( r \) is given by:
\[
A = \pi r^2
\]
2. **Proportion of the Sector**: A sector of a circle is defined by its central angle \( \theta \) (in degrees). The full circle corresponds to an angle of \( 360^\circ \).
3. **Fraction of the Circle**: The fraction of the circle that the sector represents is:
\[
\frac{\theta}{360^\circ}
\]
4. **Area of the Sector**: Therefore, the area \( A_{\text{sector}} \) of the sector can be calculated as a fraction of the area of the full circle:
\[
A_{\text{sector}} = \frac{\theta}{360^\circ} \times A = \frac{\theta}{360^\circ} \times \pi r^2
\]
From this derivation, we can see that the area of a sector is given by the formula:
\[
A_{\text{sector}} = \frac{\theta}{360^\circ} \pi r^2
\]
Looking at the options given:
- Option **b** correctly describes the process:
**b. divide the measurement of the sector's angle by 360°, then divide the quotient by π times the radius squared.**
Thus, the correct answer is **b**.
1. **Area of a Circle**: The area \( A \) of a full circle with radius \( r \) is given by:
\[
A = \pi r^2
\]
2. **Proportion of the Sector**: A sector of a circle is defined by its central angle \( \theta \) (in degrees). The full circle corresponds to an angle of \( 360^\circ \).
3. **Fraction of the Circle**: The fraction of the circle that the sector represents is:
\[
\frac{\theta}{360^\circ}
\]
4. **Area of the Sector**: Therefore, the area \( A_{\text{sector}} \) of the sector can be calculated as a fraction of the area of the full circle:
\[
A_{\text{sector}} = \frac{\theta}{360^\circ} \times A = \frac{\theta}{360^\circ} \times \pi r^2
\]
From this derivation, we can see that the area of a sector is given by the formula:
\[
A_{\text{sector}} = \frac{\theta}{360^\circ} \pi r^2
\]
Looking at the options given:
- Option **b** correctly describes the process:
**b. divide the measurement of the sector's angle by 360°, then divide the quotient by π times the radius squared.**
Thus, the correct answer is **b**.