Question

1. which of the following equations represents the area of a sector?

a = \frac{n}{360^\circ} \pi r^2, where n is the central angle of the sector
a = 360n\pi r^2, where n is the central angle of the sector
a = n\pi r^2, where n is the central angle of the sector
a = \frac{n}{360^\circ} \pi r, where n is the central angle of the sector

Explanation:

Step1: Recall Sector Area Formula

The area of a circle is \( A = \pi r^2 \). A sector is a portion of a circle, determined by the central angle \( n \) (in degrees). The fraction of the circle that the sector represents is \( \frac{n}{360^\circ} \) (since a full circle is \( 360^\circ \)). So, the area of the sector should be this fraction multiplied by the area of the circle.

Step2: Analyze Each Option

• Option 1: \( A = \frac{n}{360} \cdot \pi r^2 \). This matches the formula we derived (fraction of the circle times circle's area).
• Option 2: \( A = 360n\pi r^2 \). This would be an extremely large area (multiplying by \( 360n \)), which is incorrect for a sector (a sector is a part of the circle, so the coefficient should be a fraction less than 1 for \( n < 360 \)).
• Option 3: \( A = n\pi r^2 \). This would imply the sector is \( n \) times the circle's area, which is only true if \( n = \frac{1}{360} \), but generally, this is incorrect.
• Option 4: \( A = \frac{n}{360} \cdot \pi r \). This has a linear term \( \pi r \) (circumference - related), not area - related (which should have \( r^2 \)), so this is the formula for arc length (not area), so it's incorrect.

Answer:

A. \( A = \frac{n}{360}\cdot \pi r^2 \), where \( n \) is the central angle of the sector