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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]\( C = 2 \pi d \)[/tex] B. [tex]\( C = \pi r d \)[/tex] C. [tex]\( C = \pi d^2 \)[/tex] D. [tex]\( C = \pi d \)[/tex]
To determine the equivalent formula for the circumference [tex]\(C\)[/tex] in terms of the diameter [tex]\(d\)[/tex], we start with the known formula for the circumference of a circle:
[tex]\[ C = 2 \pi r \][/tex]
where [tex]\(r\)[/tex] is the radius of the circle. We also know the relationship between the diameter [tex]\(d\)[/tex] and the radius [tex]\(r\)[/tex] of a circle: [tex]\[ d = 2r \][/tex]
Let's substitute [tex]\(r\)[/tex] in the circumference formula using this relationship:
[tex]\[ C = 2 \pi r \][/tex] [tex]\[ r = \frac{d}{2} \][/tex]
Substitute [tex]\(\frac{d}{2}\)[/tex] for [tex]\(r\)[/tex] in the formula for circumference:
[tex]\[ C = 2 \pi \left( \frac{d}{2} \right) \][/tex]
Now, simplify the equation:
[tex]\[ C = 2 \pi \cdot \frac{d}{2} \][/tex] [tex]\[ C = \pi d \][/tex]
Therefore, the equivalent formula for the circumference [tex]\(C\)[/tex] in terms of the diameter [tex]\(d\)[/tex] is:
[tex]\[ C = \pi d \][/tex]
Looking at the given choices, the correct answer is:
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