Let's start by recalling the relationship between the radius [tex]\( r \)[/tex] and the diameter [tex]\( d \)[/tex] of a circle. The diameter [tex]\( d \)[/tex] is twice the radius [tex]\( r \)[/tex], which can be written as:
[tex]\[ d = 2r \][/tex]
The formula for the circumference [tex]\( C \)[/tex] of a circle in terms of its radius [tex]\( r \)[/tex] is:
[tex]\[ C = 2\pi r \][/tex]
Now, we need to find an equivalent formula for the circumference in terms of the diameter [tex]\( d \)[/tex]. We know from the relationship between the radius and the diameter that:
[tex]\[ r = \frac{d}{2} \][/tex]
We can substitute this expression for [tex]\( r \)[/tex] into the original circumference formula. Thus,
[tex]\[ C = 2\pi \left( \frac{d}{2} \right) \][/tex]
Next, we simplify this expression by performing the multiplication:
[tex]\[ C = 2\pi \cdot \frac{d}{2} \][/tex]
[tex]\[ C = \pi d \][/tex]
Therefore, the equivalent formula for the circumference [tex]\( C \)[/tex] in terms of the diameter [tex]\( d \)[/tex] is:
[tex]\[ C = \pi d \][/tex]
Comparing this result with the provided options, we see that the correct option is:
C. [tex]\( C = \pi d \)[/tex]
