Answer :
To determine the correct formula that relates the circumference of a circle (C) to its radius (r), we need to remember the fundamental relationship between these two quantities.
Here’s a detailed breakdown:
1. Understanding the Circumference of a Circle:
- The circumference of a circle is the total distance around the circle.
- It is directly proportional to both π (pi) and the diameter of the circle.
2. Fundamental Formula involving Diameter:
- The diameter (D) of a circle is twice the radius (r). Thus, [tex]\( D = 2r \)[/tex].
- The basic formula for the circumference [tex]\( C \)[/tex] of a circle is given by:
[tex]\[ C = \pi \times D \][/tex]
- Substituting [tex]\( D = 2r \)[/tex] into the formula, we get:
[tex]\[ C = \pi \times 2r \][/tex]
3. Simplifying the Formula:
- By rearranging, we get the well-known formula:
[tex]\[ C = 2 \pi r \][/tex]
Now, let’s compare this result with the given options:
A. [tex]\( C + 2 = \pi r \)[/tex]
- This equation does not correctly represent the relationship.
B. [tex]\( C = 2 \pi / r \)[/tex]
- This incorrectly implies division by the radius, which is not accurate.
C. [tex]\( C = 2 \pi r \)[/tex]
- This is the correct formula that we derived.
D. [tex]\( C = 2 \pi D \)[/tex]
- While it looks similar, [tex]\( D \)[/tex] stands for diameter, not radius. So, it is not the formula we are looking for in terms of radius.
Therefore, the correct answer is:
C. [tex]\( C = 2 \pi r \)[/tex]
Here’s a detailed breakdown:
1. Understanding the Circumference of a Circle:
- The circumference of a circle is the total distance around the circle.
- It is directly proportional to both π (pi) and the diameter of the circle.
2. Fundamental Formula involving Diameter:
- The diameter (D) of a circle is twice the radius (r). Thus, [tex]\( D = 2r \)[/tex].
- The basic formula for the circumference [tex]\( C \)[/tex] of a circle is given by:
[tex]\[ C = \pi \times D \][/tex]
- Substituting [tex]\( D = 2r \)[/tex] into the formula, we get:
[tex]\[ C = \pi \times 2r \][/tex]
3. Simplifying the Formula:
- By rearranging, we get the well-known formula:
[tex]\[ C = 2 \pi r \][/tex]
Now, let’s compare this result with the given options:
A. [tex]\( C + 2 = \pi r \)[/tex]
- This equation does not correctly represent the relationship.
B. [tex]\( C = 2 \pi / r \)[/tex]
- This incorrectly implies division by the radius, which is not accurate.
C. [tex]\( C = 2 \pi r \)[/tex]
- This is the correct formula that we derived.
D. [tex]\( C = 2 \pi D \)[/tex]
- While it looks similar, [tex]\( D \)[/tex] stands for diameter, not radius. So, it is not the formula we are looking for in terms of radius.
Therefore, the correct answer is:
C. [tex]\( C = 2 \pi r \)[/tex]
