If a circle's circumference is expressed using the radius, which formula is correct?

If a circle's circumference is expressed using the radius, which formula is correct?

• A. πr
• B. πd
• C. 4πr
• D. 2πr

Answer: (D)

The main idea is that the circumference scales with the radius by a factor of 2π. Starting from the relation with diameter, C = πd. Since the diameter d is twice the radius r, substitute d = 2r to get C = π(2r) = 2πr. So, when expressing circumference in terms of the radius, the correct formula is 2πr. The other forms don’t fit the requirement: πr would give half the actual circumference, missing the factor of 2. 4πr would double the correct value. πd uses diameter rather than the radius, and while it equals 2πr if you substitute d = 2r, it isn’t an expression written purely in terms of the radius.

The main idea is that the circumference scales with the radius by a factor of 2π. Starting from the relation with diameter, C = πd. Since the diameter d is twice the radius r, substitute d = 2r to get C = π(2r) = 2πr. So, when expressing circumference in terms of the radius, the correct formula is 2πr.

The other forms don’t fit the requirement: πr would give half the actual circumference, missing the factor of 2. 4πr would double the correct value. πd uses diameter rather than the radius, and while it equals 2πr if you substitute d = 2r, it isn’t an expression written purely in terms of the radius.