CONCEPT PREVIEW Find the exact length of each arc intercepted by the given central angle.

검증된 단계별 안내

Identify the given central angle \( \theta \) and the radius \( r \) of the circle. The arc length depends on these two values.

Recall the formula for the length of an arc intercepted by a central angle: \[ \text{Arc length} = r \times \theta \] where \( \theta \) must be in radians.

If the central angle \( \theta \) is given in degrees, convert it to radians using the conversion: \[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \]

Substitute the radius \( r \) and the central angle in radians \( \theta \) into the arc length formula to express the exact length of the arc.

Simplify the expression if possible, leaving the answer in terms of \( \pi \) for an exact value rather than a decimal approximation.

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주요 개념

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

Central Angle

A central angle is an angle whose vertex is at the center of a circle and whose sides intersect the circle, creating an arc. Understanding the measure of this angle is essential because it directly relates to the length of the intercepted arc.

Arc Length Formula

The arc length of a circle segment is calculated using the formula: Arc Length = (θ/360) × 2πr, where θ is the central angle in degrees and r is the radius. This formula connects the angle measure to the portion of the circle's circumference.

Radians and Degrees Conversion

Angles can be measured in degrees or radians, and converting between these units is often necessary. Since the arc length formula can use radians (Arc Length = rθ), knowing how to convert degrees to radians (radians = degrees × π/180) is crucial for accurate calculations.