To find the length of a 180° arc in a circle with a diameter of 16 feet, follow these steps:
1. Find the radius of the circle:
The radius \( r \) is half of the diameter \( d \).
[tex]\[
r = \frac{d}{2} = \frac{16 \text{ feet}}{2} = 8 \text{ feet}
\][/tex]
2. Calculate the circumference of the circle:
The circumference \( C \) of a circle is given by the formula:
[tex]\[
C = \pi \times d
\][/tex]
Substituting the given diameter:
[tex]\[
C = \pi \times 16 \text{ feet}
\][/tex]
3. Determine the proportion of the circumference for the 180° arc:
Since 180° is exactly half of a full circle (which is 360°), the 180° arc is half of the circumference of the circle. The length of the arc \( L \) can be found using the proportion of the angle to the full circle:
[tex]\[
L = \left(\frac{180^\circ}{360^\circ}\right) \times C = \left(\frac{1}{2}\right) \times \pi \times 16 \text{ feet}
\][/tex]
Simplifying this:
[tex]\[
L = \frac{1}{2} \times 16\pi \text{ feet} = 8\pi \text{ feet}
\][/tex]
Therefore, the exact length of the 180° arc in simplest form is:
[tex]\[
8\pi \text{ feet}
\][/tex]
