Math101 · Independent Ontario learning library
Math101learn.math101.caCircumference
Circumference is the distance around a circle. It equals $C=2\pi r$ or $C=\pi d$, where $d=2r$.
Circumference measures borders, wheel travel, belts, pipes, and circular tracks. Its direct proportionality to diameter defines $\pi$.
Intuition and core definition
Circumference is the distance around a circle. It equals $C=2\pi r$ or $C=\pi d$, where $d=2r$. The constant $\pi$ is the ratio of circumference to diameter for every circle, so it measures a universal scaling relationship.
Notation, language, and conditions
$C$ has linear units. An exact value retains $\pi$; an approximation uses $\approx$ and a stated rounding level. Circumference differs from disk area $\pi r^2$ and from an arc length, which is only a fraction of the full boundary.
Why this idea matters
Circumference measures boundary length and is proportional to diameter through the constant $\pi$.
A dependable method
- Determine whether radius or diameter is given.
- Use $2\pi r$ for radius or $\pi d$ for diameter.
- Keep units consistent and calculate an exact form.
- Approximate with a calculator only if requested.
- Compare with roughly three diameters because $\pi\approx3.14$.
Worked example
Representations and interpretation
Wrapping string once around a circle and comparing it with the diameter shows a length a little more than three diameters. Unrolling the boundary turns circumference into a measurable straight segment.
Reasoning about variations
If radius increases by $3$ units, circumference increases by $2\pi(3)=6\pi$ units regardless of starting radius. This constant rate contrasts with area’s nonconstant increase.
Common mistakes
How to check your work
- Divide circumference by diameter and expect approximately $\pi$.
- Estimate as slightly more than three diameters.
- Reverse with $r=C/(2\pi)$ or $d=C/\pi$.
Practice
- Find the circumference of radius $7$ cm.
- A circle has circumference $12\pi$ m. Find its diameter.
- A radius triples. By what factor does circumference change?
Answers and brief solutions
Show answers
- $14\pi$ cm $C=2\pi(7)$.
- $12$ m $C=\pi d$.
- $3$ Circumference is directly proportional to radius.
Synthesis and transfer
One wheel revolution advances approximately one circumference, so counting revolutions can estimate travel distance and units remain linear rather than squared.
A bicycle wheel with radius $0.34$ m travels $2\pi(0.34)$ metres per full revolution if it rolls without slipping. Multiplying that length by the revolution count estimates total distance; conversely, dividing a measured route by circumference estimates rotations. Tire deformation and slipping explain why a real odometer may require calibration, even when the geometric model is sound. Diameter enters linearly, so doubling it doubles distance per revolution. The answer uses metres, not square metres, and an estimate just above twice the diameter gives a quick check on any calculated circumference.
Related topics
Teaching and accessibility note
Explore the idea
Geometry measurement
Change one quantity at a time and connect what moves to Circumference.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find the circumference of radius $7$ cm.
- $C=2\pi(7)$.
End of lesson
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