Question

The formula for the circumference of a circle is C = πd
𝐶

=

𝜋
𝑑
, where C = circumference and d = diameter. It can also be defined as C=2πr
𝐶
=
2
𝜋
𝑟
, where r = radius. If 360 degrees = 2π
2
𝜋
radians, and the formula for arc length is θ=sr,
𝜃
=
𝑠
𝑟
,

the determine the length created by a 30 degree central angle in a circle with a 2 inch diameter and explain how you found it. Leave your answer in terms of pi. You may type out your work or upload it.

Answer 1

To determine the length of the arc created by a 30-degree central angle in a circle with a 2-inch diameter, we can use the formula for the arc length:

\[
s = r \theta
\]

Where:
- \(s\) is the arc length,
- \(r\) is the radius of the circle,
- \(\theta\) is the central angle in radians.

### Step 1: Find the radius

The diameter \(d\) of the circle is given as 2 inches. The radius \(r\) is half of the diameter:

\[
r = \frac{d}{2} = \frac{2 \text{ inches}}{2} = 1 \text{ inch}
\]

### Step 2: Convert the angle from degrees to radians

We need to convert the angle from degrees to radians. The conversion is done using the fact that \(360\) degrees is equal to \(2\pi\) radians:

\[
\theta = 30^\circ \times \left( \frac{2\pi \text{ radians}}{360^\circ} \right) = 30^\circ \times \left( \frac{\pi}{180} \right) = \frac{30\pi}{180} = \frac{\pi}{6} \text{ radians}
\]

### Step 3: Calculate the arc length using the formula

Now, substitute \(r\) and \(\theta\) into the arc length formula:

\[
s = r \theta = 1 \text{ inch} \times \frac{\pi}{6} \text{ radians} = \frac{\pi}{6} \text{ inches}
\]

### Conclusion

The length of the arc created by a 30 degree central angle in a circle with a 2-inch diameter is

\[
s = \frac{\pi}{6} \text{ inches}.
\]