A student is asked to find the circumference of a circle with radius 6 cm. She uses the formula C = π × d and substitutes 6 for d, getting C = 6π cm. Is her answer correct?
AYes — both formulas give the same answer, so substituting either r or d produces 6π
BNo — she substituted the radius where the formula requires the diameter; the correct answer is 12π cm
CNo — she should have used C = 2πr, and that formula gives C = 36π cm
DYes — for a circle with radius 6, the diameter is also 6
C = πd requires the diameter, not the radius. The diameter is twice the radius, so d = 2 × 6 = 12 cm. The correct answer is C = π × 12 = 12π cm. Alternatively, using C = 2πr directly: C = 2 × π × 6 = 12π cm. Both formulas give the same answer because they express the same relationship — but you must use the right measurement in each formula. The most common error is plugging the radius into C = πd.
Question 2 Multiple Choice
You measure the circumference and diameter of three circular objects: a coin, a dinner plate, and a bicycle wheel. You then compute C ÷ d for each. What should you find?
AA different ratio for each object, since larger circles have a larger ratio
BA ratio close to 3.14159 for each object, because this ratio is constant for all circles
CA ratio equal to the radius of each circle
DA ratio of exactly 3.14 for each, because π equals 3.14
Pi is defined as the ratio C/d, and this ratio is the same for every circle regardless of size. This is a discovered fact of geometry, not a human convention. A coin and a bicycle wheel have vastly different sizes but identical C/d ratios — approximately 3.14159. Note that 3.14 is only an approximation; the true value of π is irrational and never terminates or repeats exactly. This universality is what makes π a fundamental constant rather than a circle-specific measurement.
Question 3 True / False
The ratio of circumference to diameter is the same for every circle, regardless of its size.
TTrue
FFalse
Answer: True
This is the fundamental property that defines π. No matter how large or small the circle, C/d = π ≈ 3.14159. This can be verified empirically by measuring real circular objects and computing the ratio — it is always approximately the same value. Pi is a fixed constant of geometry, not a variable that depends on circle size. This is why it appears in both circumference formulas: the ratio is always π, so C = π × d.
Question 4 True / False
The circumference of a circle with radius 5 cm is 5π cm.
TTrue
FFalse
Answer: False
The circumference is C = 2πr = 2 × π × 5 = 10π cm. Equivalently, the diameter is d = 2r = 10 cm, so C = πd = 10π cm. The answer 5π would result from forgetting to multiply by 2 — either using C = πr (incorrect formula) or using the radius where the diameter belongs. This is the most common circumference error: confusing radius and diameter in the formula.
Question 5 Short Answer
Explain why the formulas C = πd and C = 2πr give identical results for any circle. What is the relationship that makes them equivalent?
Think about your answer, then reveal below.
Model answer: The diameter of a circle is always exactly twice its radius: d = 2r. Substituting this into C = πd gives C = π(2r) = 2πr, which is exactly the other formula. They express the same relationship in terms of different measurements of the same circle. If you know the radius, use C = 2πr. If you know the diameter, use C = πd. Both arrive at the same circumference because the two formulas are algebraically identical.
The equivalence is algebraic, not coincidental. The underlying truth is C/d = π for every circle, which gives C = πd. Since d = 2r always, substituting yields C = 2πr. Students sometimes think the two formulas are different rules — they are one rule expressed two ways depending on which circle measurement you start with.