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?

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



Answer :

To determine which of the given options is an equivalent formula for the circumference of a circle, we start with the known relationships and formulas.

1. Circumference Formula:
The circumference [tex]\( C \)[/tex] of a circle in terms of its radius [tex]\( r \)[/tex] is given by:
[tex]\[ C = 2 \pi r \][/tex]

2. Relationship Between Diameter and Radius:
The diameter [tex]\( d \)[/tex] of a circle is twice the radius [tex]\( r \)[/tex]. Mathematically, this can be written as:
[tex]\[ d = 2r \][/tex]

3. Substituting Diameter into Circumference Formula:
We can express the circumference formula in terms of the diameter [tex]\( d \)[/tex] by substituting [tex]\( d = 2r \)[/tex] into [tex]\( C = 2 \pi r \)[/tex]. First, solve that for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{d}{2} \][/tex]
Now substitute [tex]\( r = \frac{d}{2} \)[/tex] back into the original circumference formula:
[tex]\[ C = 2 \pi \left( \frac{d}{2} \right) \][/tex]

4. Simplifying the Expression:
Simplify the expression inside the parentheses:
[tex]\[ C = 2 \pi \cdot \frac{d}{2} \][/tex]

The [tex]\( 2 \)[/tex] in the numerator and the [tex]\( 2 \)[/tex] in the denominator cancel out:
[tex]\[ C = \pi d \][/tex]

Hence, the equivalent formula for the circumference [tex]\( C \)[/tex] using the diameter [tex]\( d \)[/tex] is:
[tex]\[ C = \pi d \][/tex]

After reviewing the given options, we find the correct choice:
[tex]\[ \boxed{C = \pi d} \][/tex]

So, the correct answer is option [tex]\( \text{C} \)[/tex]: [tex]\( C = \pi d \)[/tex].