Question

1. explain in your own words why the arc length formula includes the fraction $\theta/360\degree$.
2. how are the formulas for arc length and sector area similar? how are they different?

Explanation:

Question 17:

Step 1: Recall Circle Basics

A full circle has \( 360^\circ \) and its circumference is \( C = 2\pi r \) (or \( C=\pi d \), where \( d \) is diameter).

Step 2: Arc as Part of Circumference

An arc is a portion of the circle’s circumference. The fraction \( \frac{\theta}{360^\circ} \) represents what part of the full \( 360^\circ \) circle the central angle \( \theta \) (defining the arc) occupies.

Step 3: Relate to Arc Length

To find the arc length (\( L \)), we take the fraction of the full circumference corresponding to \( \theta \). So:
\( L = \frac{\theta}{360^\circ} \times 2\pi r \) (or \( L = \frac{\theta}{360^\circ} \times \pi d \)). The \( \frac{\theta}{360^\circ} \) scales the full circumference to the length of the arc defined by \( \theta \).

Step 1: Recall Formulas

• Arc Length (\( L \)): \( L = \frac{\theta}{360^\circ} \times 2\pi r \) (or \( \frac{\theta}{360^\circ} \times \pi d \)).
• Sector Area (\( A \)): \( A = \frac{\theta}{360^\circ} \times \pi r^2 \).

Step 2: Identify Similarities

Both formulas use \( \frac{\theta}{360^\circ} \) to represent the fraction of the full circle (by central angle \( \theta \)) that the arc (for length) or sector (for area) occupies.

Step 3: Identify Differences

• Arc Length relates to the circle’s perimeter (circumference). It scales the full circumference (\( 2\pi r \)) by \( \frac{\theta}{360^\circ} \).
• Sector Area relates to the circle’s area (\( \pi r^2 \)). It scales the full area by \( \frac{\theta}{360^\circ} \).

Answer:

The arc length formula includes \( \frac{\theta}{360^\circ} \) because a circle has \( 360^\circ \), and \( \theta \) is the central angle of the arc. The fraction \( \frac{\theta}{360^\circ} \) represents the proportion of the full circle’s circumference (which is \( 2\pi r \)) that the arc occupies. Multiplying the full circumference by this fraction gives the length of the arc (since the arc is a “slice” of the circle’s perimeter corresponding to angle \( \theta \)).

Question 18: