To determine which of the given options is an equivalent formula for the circumference of a circle, we start with the known relationships and formulas.
1. Circumference Formula:
The circumference [tex]\( C \)[/tex] of a circle in terms of its radius [tex]\( r \)[/tex] is given by:
[tex]\[
C = 2 \pi r
\][/tex]
2. Relationship Between Diameter and Radius:
The diameter [tex]\( d \)[/tex] of a circle is twice the radius [tex]\( r \)[/tex]. Mathematically, this can be written as:
[tex]\[
d = 2r
\][/tex]
3. Substituting Diameter into Circumference Formula:
We can express the circumference formula in terms of the diameter [tex]\( d \)[/tex] by substituting [tex]\( d = 2r \)[/tex] into [tex]\( C = 2 \pi r \)[/tex]. First, solve that for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{d}{2}
\][/tex]
Now substitute [tex]\( r = \frac{d}{2} \)[/tex] back into the original circumference formula:
[tex]\[
C = 2 \pi \left( \frac{d}{2} \right)
\][/tex]
4. Simplifying the Expression:
Simplify the expression inside the parentheses:
[tex]\[
C = 2 \pi \cdot \frac{d}{2}
\][/tex]
The [tex]\( 2 \)[/tex] in the numerator and the [tex]\( 2 \)[/tex] in the denominator cancel out:
[tex]\[
C = \pi d
\][/tex]
Hence, the equivalent formula for the circumference [tex]\( C \)[/tex] using the diameter [tex]\( d \)[/tex] is:
[tex]\[
C = \pi d
\][/tex]
After reviewing the given options, we find the correct choice:
[tex]\[
\boxed{C = \pi d}
\][/tex]
So, the correct answer is option [tex]\( \text{C} \)[/tex]: [tex]\( C = \pi d \)[/tex].
