Question

The formula for the circumference of a circle is
C

=

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

the determine the arc 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

Answer 1

To find the arc length created by a 30-degree central angle in a circle with a 2-inch diameter, we can follow these steps:

1. **Calculate the radius**:
The radius \( r \) is half of the diameter. Given that the diameter \( d = 2 \) inches, we calculate the radius as:
\[
r = \frac{d}{2} = \frac{2 \text{ inches}}{2} = 1 \text{ inch}
\]

2. **Convert degrees to radians**:
Since the formula for arc length involves radians, we need to convert the 30 degrees to radians. We know that:
\[
360 \text{ degrees} = 2\pi \text{ radians}
\]
Therefore, to convert degrees to radians, we use the following conversion factor:
\[
\text{radians} = \frac{\text{degrees}}{360} \times 2\pi
\]
Applying this to 30 degrees:
\[
\text{radians} = \frac{30}{360} \times 2\pi = \frac{1}{12} \times 2\pi = \frac{\pi}{6}
\]

3. **Use the arc length formula**:
The formula for arc length \( s \) in terms of the central angle in radians is:
\[
s = \theta r
\]
Where \( \theta \) is the central angle in radians, and \( r \) is the radius. Substituting the values we have:
\[
s = \left(\frac{\pi}{6}\right)(1 \text{ inch})
\]

4. **Calculate the arc length**:
Now we can find the arc length:
\[
s = \frac{\pi}{6} \text{ inches}
\]

Thus, the arc length created by a 30-degree central angle in a circle with a 2-inch diameter is:
\[
\boxed{\frac{\pi}{6}} \text{ inches}
\]