Answer :
To solve the problem of finding the equivalent formula for the circumference of a circle given the options, we first need to understand the basic relationships involving the radius [tex]\( r \)[/tex], diameter [tex]\( d \)[/tex], and circumference [tex]\( C \)[/tex].
1. Recall the Formula for Circumference:
The circumference [tex]\( C \)[/tex] of a circle is given by:
[tex]\[ C = 2\pi r \][/tex]
2. Relationship Between Radius and Diameter:
The diameter [tex]\( d \)[/tex] of a circle is twice the radius:
[tex]\[ d = 2r \][/tex]
3. Rewriting Circumference in Terms of Diameter:
Substitute [tex]\( d = 2r \)[/tex] into the circumference formula:
[tex]\[ C = 2\pi r = \pi d \][/tex]
This tells us that the circumference can also be expressed as:
[tex]\[ C=\pi d \][/tex]
4. Examine the Options:
- Option A: [tex]\( C = \pi d \)[/tex]
This matches our derived formula exactly, so this is correct.
- Option B: [tex]\( C = \pi d r \)[/tex]
This formulation does not match the derived formula. To check this, if we plug in specific values where [tex]\( r = 1 \)[/tex] and [tex]\( d = 2r = 2 \)[/tex]:
[tex]\[ C = \pi \cdot 2 \cdot 1 = 2\pi, \][/tex]
but [tex]\( C = 2\pi r = \pi d \)[/tex]. This is incorrect since it introduces an unnecessary multiplication by [tex]\( r \)[/tex].
- Option C: [tex]\( C = \pi r^2 \)[/tex]
This is the formula for the area of a circle, not the circumference. Hence, this is incorrect. For [tex]\( r = 1 \)[/tex]:
[tex]\[ \pi r^2 = \pi \cdot 1^2 = \pi \][/tex]
giving us the area, not the circumference.
- Option D: [tex]\( C = 2 \pi d \)[/tex]
Using [tex]\( d = 2r \)[/tex], we get:
[tex]\[ C = 2 \pi \cdot 2r = 4 \pi r \][/tex]
which does not simplify into the form [tex]\( 2 \pi r \)[/tex]. Therefore, this is also incorrect.
5. Verify with Given Values:
By evaluating with a radius of [tex]\( r = 1 \)[/tex] unit, we calculate the diameter:
[tex]\[ d = 2 \times 1 = 2 \][/tex]
Calculate the actual circumference which is:
[tex]\[ C = 2 \pi \cdot 1 = 2\pi \approx 6.2831853072 \][/tex]
Check each option:
- [tex]\( A = \pi \cdot 2 \approx 6.2831853072 \)[/tex]
- [tex]\( B = \pi \cdot 2 \cdot 1 \approx 6.2831853072 \)[/tex]
- [tex]\( C = \pi \cdot 1^2 \approx 3.1415926536 \)[/tex]
- [tex]\( D = 2 \pi \cdot 2 \approx 12.5663706144 \)[/tex]
This confirms the correct option is:
Option A: [tex]\( C = \pi d \)[/tex]
1. Recall the Formula for Circumference:
The circumference [tex]\( C \)[/tex] of a circle is given by:
[tex]\[ C = 2\pi r \][/tex]
2. Relationship Between Radius and Diameter:
The diameter [tex]\( d \)[/tex] of a circle is twice the radius:
[tex]\[ d = 2r \][/tex]
3. Rewriting Circumference in Terms of Diameter:
Substitute [tex]\( d = 2r \)[/tex] into the circumference formula:
[tex]\[ C = 2\pi r = \pi d \][/tex]
This tells us that the circumference can also be expressed as:
[tex]\[ C=\pi d \][/tex]
4. Examine the Options:
- Option A: [tex]\( C = \pi d \)[/tex]
This matches our derived formula exactly, so this is correct.
- Option B: [tex]\( C = \pi d r \)[/tex]
This formulation does not match the derived formula. To check this, if we plug in specific values where [tex]\( r = 1 \)[/tex] and [tex]\( d = 2r = 2 \)[/tex]:
[tex]\[ C = \pi \cdot 2 \cdot 1 = 2\pi, \][/tex]
but [tex]\( C = 2\pi r = \pi d \)[/tex]. This is incorrect since it introduces an unnecessary multiplication by [tex]\( r \)[/tex].
- Option C: [tex]\( C = \pi r^2 \)[/tex]
This is the formula for the area of a circle, not the circumference. Hence, this is incorrect. For [tex]\( r = 1 \)[/tex]:
[tex]\[ \pi r^2 = \pi \cdot 1^2 = \pi \][/tex]
giving us the area, not the circumference.
- Option D: [tex]\( C = 2 \pi d \)[/tex]
Using [tex]\( d = 2r \)[/tex], we get:
[tex]\[ C = 2 \pi \cdot 2r = 4 \pi r \][/tex]
which does not simplify into the form [tex]\( 2 \pi r \)[/tex]. Therefore, this is also incorrect.
5. Verify with Given Values:
By evaluating with a radius of [tex]\( r = 1 \)[/tex] unit, we calculate the diameter:
[tex]\[ d = 2 \times 1 = 2 \][/tex]
Calculate the actual circumference which is:
[tex]\[ C = 2 \pi \cdot 1 = 2\pi \approx 6.2831853072 \][/tex]
Check each option:
- [tex]\( A = \pi \cdot 2 \approx 6.2831853072 \)[/tex]
- [tex]\( B = \pi \cdot 2 \cdot 1 \approx 6.2831853072 \)[/tex]
- [tex]\( C = \pi \cdot 1^2 \approx 3.1415926536 \)[/tex]
- [tex]\( D = 2 \pi \cdot 2 \approx 12.5663706144 \)[/tex]
This confirms the correct option is:
Option A: [tex]\( C = \pi d \)[/tex]
