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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]\( C = \pi d \)[/tex]
C. [tex]\( C = \pi dr \)[/tex]
D. [tex]\( C = \pi r^2 \)[/tex]

Sagot :

GinnyAnsweridn
To determine which of the given formulas is equivalent to the circumference [tex]\( C = 2 \pi r \)[/tex] of a circle with radius [tex]\( r \)[/tex], let's go through the relationships step-by-step.

### Step 1: Understanding the Relationships

1. Radius and Diameter:
- The diameter [tex]\( d \)[/tex] of a circle is twice the radius [tex]\( r \)[/tex], i.e., [tex]\( d = 2r \)[/tex].

2. Circumference of a Circle:
- The standard formula for the circumference of a circle in terms of its radius [tex]\( r \)[/tex] is [tex]\( C = 2 \pi r \)[/tex].

### Step 2: Substituting [tex]\( d = 2r \)[/tex] into the Circumference Formula
To express the circumference in terms of the diameter [tex]\( d \)[/tex]:

1. Start with the formula for circumference:
[tex]\[ C = 2 \pi r \][/tex]

2. Substitute [tex]\( r = \frac{d}{2} \)[/tex] (since [tex]\( d = 2r \)[/tex]):
[tex]\[ C = 2 \pi \left(\frac{d}{2}\right) \][/tex]

3. Simplify:
[tex]\[ C = 2 \pi \cdot \frac{d}{2} \][/tex]
[tex]\[ C = \pi d \][/tex]

### Conclusion
The formula [tex]\( C = \pi d \)[/tex] is equivalent to the given formula [tex]\( C = 2\pi r \)[/tex].

Thus, the correct answer from the given options is:
B. [tex]\( C = \pi d \)[/tex]
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