Answered

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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]\( C = \pi d^2 \)[/tex]
B. [tex]\( C = \pi d \)[/tex]
C. [tex]\( C = \pi rd \)[/tex]
D. [tex]\( C = 2\pi d \)[/tex]


Sagot :

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]