To solve this problem, we start by recalling the standard formula for the circumference of a circle, which is given by
[tex]\[ C = 2\pi r \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle. We also know that the diameter [tex]\( d \)[/tex] of the circle is related to the radius by the formula:
[tex]\[ d = 2r \][/tex]
Our goal is to express the circumference [tex]\( C \)[/tex] in terms of the diameter [tex]\( d \)[/tex].
First, substitute the expression for the diameter [tex]\( d \)[/tex] into the circumference formula:
[tex]\[ C = 2\pi r \][/tex]
Using the relationship [tex]\( d = 2r \)[/tex], we can solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{d}{2} \][/tex]
Now substitute this expression for [tex]\( r \)[/tex] back into the formula for the circumference:
[tex]\[ C = 2\pi \left(\frac{d}{2}\right) \][/tex]
Simplifying this equation, we get:
[tex]\[ C = 2\pi \cdot \frac{d}{2} \][/tex]
[tex]\[ C = \pi d \][/tex]
Thus, the equivalent formula for the circumference in terms of the diameter is
[tex]\[ C = \pi d \][/tex]
From the given options, the correct one aligns with:
C. [tex]\( C = \pi d \)[/tex]
Therefore, the answer is option C.
