To determine which formula is equivalent to the circumference formula [tex]\(C=2\pi r\)[/tex] using the diameter, we start by understanding the relationship between the radius [tex]\(r\)[/tex] and the diameter [tex]\(d\)[/tex].
### Step-by-Step Solution:
1. Identify the relationships:
- The diameter [tex]\(d\)[/tex] of a circle is twice the radius [tex]\(r\)[/tex].
[tex]\[
d = 2r
\][/tex]
2. Substitute the radius [tex]\(r\)[/tex] in the circumference formula:
- The original formula for the circumference is [tex]\(C = 2\pi r\)[/tex].
- Substitute [tex]\(r\)[/tex] with [tex]\(\frac{d}{2}\)[/tex] since [tex]\(d = 2r\)[/tex]:
[tex]\[
C = 2 \pi \left(\frac{d}{2}\right)
\][/tex]
3. Simplify the expression:
- Multiplying [tex]\(2\)[/tex] and [tex]\(\frac{d}{2}\)[/tex]:
[tex]\[
C = 2 \pi \frac{d}{2} = \pi d
\][/tex]
Thus, the equivalent formula for the circumference [tex]\(C\)[/tex] in terms of the diameter [tex]\(d\)[/tex] is:
[tex]\[
C = \pi d
\][/tex]
### Evaluate the Given Options:
- Option A: [tex]\(C = \pi d^2\)[/tex]
- This involves squaring the diameter, which is not justifiable from the given circumference formula.
- Option B: [tex]\(C = \pi d\)[/tex]
- This matches the simplified equivalent expression [tex]\(C = \pi d\)[/tex].
- Option C: [tex]\(C = \pi r d\)[/tex]
- This mixes radius and diameter incorrectly and does not simplify to the given formula.
- Option D: [tex]\(C = 2 \pi d\)[/tex]
- This doubles the diameter incorrectly; the formula does not match [tex]\(C = 2 \pi r\)[/tex].
### Conclusion:
- The correct answer is:
[tex]\[
\boxed{B}
\][/tex]
