Talia's understanding is on the right track, but it's important to clarify the connection between the circumference of a circle and the arc length. The formula for the circumference of a circle is given as \( C = 2\pi r \) when using the radius \( r \). When measuring an arc length in radians, we use the formula \( L = r\theta \), where \( L \) is the arc length and \( \theta \) is the angle in radians. If the angle \( \theta \) is \( 2\pi \) radians, which corresponds to a full circle, then the arc length becomes the entire circumference, confirming the relationship between the two formulas. Thus, the circumference formula can indeed be applied to find arc lengths for specific angles in radians.
Talia has been learning about arc lengths in geometry. She comes to class one day and says she has discovered that the formula for circumference can be converted to an arc length (in radians) formula by changing C=πd to C=2πr . In 3–5 sentences, explain how the formula for circumference can be defined as an arc length in radians.(4 points)
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