91Ó°ÊÓ

Arc length is a measurement that tells us the distance between two points along a circle. Imagine unfolding a section of the outer circle onto a straight line, the arc length would be the length of that line.
The formula for calculating the arc length of a circle involves two key elements:

• Radius \( r \): The distance from the center of the circle to any point on its perimeter.
• Central angle \( \theta \): The angle subtended by the arc at the center of the circle, measured in radians.

The formula is given by: \[ s = r \cdot \theta \]where \( s \) represents the arc length.
It’s essential to ensure \( \theta \) is in radians, not degrees, when you apply this formula. Using this, you can easily calculate the arc's length if both the radius and central angle are known.

The central angle plays a critical role in determining the arc length. The central angle is the angle formed at the center of the circle by two radii.
This angle opens up to intersect the boundary of the circle, creating an arc. Central angles are usually measured in radians rather than degrees for ease of calculation in mathematics involving circles.
A full circle encompasses a central angle of \(2\pi\) radians, corresponding to \(360^\circ\). Understanding the central angle allows one to accurately determine and calculate the arc length using its relationship with the radius in the formula: \[ s = r \cdot \theta \]
Always remember, the central angle directly impacts the arc length; larger angles result in longer arcs and vice versa.

Finding the radius of a circle, given the arc length and central angle, involves rearranging the arc length formula. This process is straightforward:

• Start with the general arc length formula: \ s = r \cdot \theta \.
• Solve for the radius: \ r = \frac{s}{\theta} \.

This rearrangement allows you to find the radius by dividing the arc length by the central angle.
In the given problem, with an arc length \( s = 4.2 \) inches and central angle \( \theta = 1.4 \), the computation is straightforward:\[ r = \frac{4.2}{1.4} = 3 \text{ inches} \]
By knowing the arc length and central angle, this formula simplifies the process of determining the radius, which is a fundamental concept in circle geometry.