Certainly! Let's analyze the formulas given in the multiple-choice question, step by step, to determine which is equivalent to the circumference formula [tex]\(C = 2 \pi r\)[/tex].
1. Understand the relationship between the radius [tex]\(r\)[/tex] and the diameter [tex]\(d\)[/tex]:
The diameter [tex]\(d\)[/tex] of a circle is twice the radius [tex]\(r\)[/tex]. Mathematically, this relationship is given by:
[tex]\[
d = 2r
\][/tex]
2. Substitute [tex]\(d\)[/tex] into the original circumference formula:
The original formula for the circumference of a circle is:
[tex]\[
C = 2 \pi r
\][/tex]
Since [tex]\(d = 2r\)[/tex], we can substitute [tex]\(r = \frac{d}{2}\)[/tex] into the formula for circumference.
3. Simplify the expression:
Substitute [tex]\(r\)[/tex] in the circumference formula:
[tex]\[
C = 2 \pi \left(\frac{d}{2}\right)
\][/tex]
Here, simplify the multiplication inside the parentheses:
[tex]\[
C = 2 \pi \cdot \frac{d}{2}
\][/tex]
The 2 in the numerator and denominator cancels out, leaving:
[tex]\[
C = \pi d
\][/tex]
4. Identify the equivalent formula:
This shows that the circumference [tex]\(C\)[/tex] can also be expressed as:
[tex]\[
C = \pi d
\][/tex]
Therefore, the equivalent formula to [tex]\(C = 2 \pi r\)[/tex] is:
[tex]\[
\boxed{C = \pi d}
\][/tex]
Thus, the correct answer is:
[tex]\[
A. \; C = \pi d
\][/tex]
