To determine the length of a chord in a circle, you can use the following formula:

In this formula, $r$ represents the radius of the circle, and $\theta$ is the central angle subtended by the chord at the center of the circle, measured in radians. This relationship is derived from trigonometry, specifically utilizing the sine function, which relates the angle to the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.

Here’s a step-by-step guide on how to apply this formula:

1. Measure or obtain the radius $r$ of the circle.
2. Identify the central angle $\theta$ that the chord subtends at the center of the circle. Ensure that this angle is expressed in radians. If the angle is provided in degrees, convert it to radians using the conversion formula: $$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$$
3. Compute $\theta/2$ by dividing the central angle $\theta$ by 2.
4. Calculate the sine of $\theta/2$.
5. Finally, multiply the sine value by 2 and then by the radius $r$ to find the chord length.

Example Calculation:

Consider a circle with a radius of $5 \, \text{cm}$ and a central angle of $60^\circ$. First, convert the angle from degrees to radians:

Next, compute $\theta/2$:

Now, find the sine of $\frac{\pi}{6}$:

Finally, substitute these values into the chord length formula:

Thus, the length of the chord is $5 \, \text{cm}$.