To determine which formula is equivalent to [tex]\( C = 2\pi r \)[/tex], let's explore the relationship between the radius [tex]\( r \)[/tex] and the diameter [tex]\( d \)[/tex] of a circle.
1. Understand the relationship between radius and diameter:
[tex]\[
d = 2r
\][/tex]
The diameter [tex]\( d \)[/tex] is twice the radius [tex]\( r \)[/tex].
2. Substitute [tex]\( d = 2r \)[/tex] into the circumference formula:
[tex]\[
C = 2\pi r
\][/tex]
We need to express [tex]\( C \)[/tex] in terms of [tex]\( d \)[/tex].
3. Replace [tex]\( r \)[/tex] with [tex]\( \frac{d}{2} \)[/tex]:
[tex]\[
C = 2\pi \left(\frac{d}{2}\right)
\][/tex]
4. Simplify the expression:
[tex]\[
C = 2\pi \cdot \frac{d}{2}
\][/tex]
[tex]\[
C = \pi d
\][/tex]
So, the equivalent formula for [tex]\( C = 2\pi r \)[/tex] when expressed in terms of the diameter [tex]\( d \)[/tex] is [tex]\( C = \pi d \)[/tex].
The correct choice from the given options is:
B. [tex]\( C = \pi d \)[/tex]
