CONCEPT PREVIEW Find the area of each sector.

검증된 단계별 안내

Identify the given information for the sector: the radius \(r\) of the circle and the central angle \(\theta\) of the sector. The angle should be in degrees or radians.

Recall the formula for the area of a sector: \(\text{Area} = \frac{\theta}{360} \times \pi r^{2}\) if \(\theta\) is in degrees, or \(\text{Area} = \frac{1}{2} r^{2} \theta\) if \(\theta\) is in radians.

If the angle \(\theta\) is given in degrees, use the first formula. If it is in radians, use the second formula. Convert the angle to the appropriate unit if necessary.

Substitute the values of \(r\) and \(\theta\) into the chosen formula to set up the expression for the area of the sector.

Simplify the expression to find the area of the sector (do not calculate the final numeric value unless asked).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definition of a Sector

A sector is a portion of a circle enclosed by two radii and the arc between them. Understanding what constitutes a sector helps in visualizing the problem and identifying the relevant parts of the circle needed to calculate the area.

Formula for the Area of a Sector

The area of a sector is given by (θ/360) × πr², where θ is the central angle in degrees and r is the radius of the circle. This formula relates the fraction of the circle represented by the sector to the total area of the circle.

Conversion Between Radians and Degrees

Since angles can be given in radians or degrees, knowing how to convert between these units (1 radian = 180/π degrees) is essential. This ensures the correct application of the area formula depending on the angle's unit.