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If [tex]$r$[/tex] is the radius of a circle and [tex]$d$[/tex] is its diameter, which of the following is an equivalent formula for the circumference [tex]$C=2 \pi r$[/tex]?

A. [tex][tex]$C=\pi d^2$[/tex][/tex]
B. [tex]$C=\pi d$[/tex]
C. [tex]$C=\pi r d$[/tex]
D. [tex][tex]$C=2 \pi d$[/tex][/tex]



Answer :

GinnyAnswer
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]

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