Circumference of a Circle

How long is the fence around a circular field? Wrap a tape measure once around the boundary, the length you record is the circumference. The remarkable fact is that the ratio $C/d$ is the same number for every circle: the constant $\pi$. Multiply the diameter by $\pi$ and you have the boundary length, no matter how big or small the circle is.

The formula

For a circle of radius $r$ (so diameter $d = 2r$):

$C = 2\pi r \;=\; \pi d.$

This is a direct rewrite of the definition $\pi = C/d$. Memorise either form, they are the same equation in two costumes.

• Mechanical Engineering: The length of belt needed to wrap around a pulley of radius $r$ is the circumference $2\pi r$ (plus the straight runs); machinists use this every time they order replacement timing belts.
• Geodesy: The Earth's circumference of approximately 40,000 km was first computed by Eratosthenes around 240 BC using $C = 2\pi r$ and a measured arc; his estimate was within 2% of the modern value.
• Physics: Centripetal acceleration $a = v^2/r$ for a body moving at constant speed around a circle of radius $r$ requires knowing the period $T = 2\pi r / v$, so the circumference formula is the entry point to all circular-motion problems.

Inscribe a regular $n$-gon inside a circle of radius $r$. The polygon's perimeter is the dashed orange outline; as you crank up $n$ the polygon hugs the circle ever more closely and the perimeter readout converges to the true circumference $2\pi r$. The widget also displays the implied estimate of $\pi$, this is the trick Archimedes used in 250 BCE.

The elegant link between area and circumference

Look at the two formulas side by side: $A(r) = \pi r^2$ and $C(r) = 2\pi r$. The circumference is exactly the derivative of the area with respect to the radius:

$\frac{dA}{dr} = 2\pi r = C.$

This is not a coincidence. If you increase $r$ by a tiny amount $\Delta r$, you add a thin ring around the outside of the disc. The ring has length $C$ and width $\Delta r$, so its area is approximately $C \cdot \Delta r$. Dividing by $\Delta r$ recovers the circumference. This geometric intuition foreshadows the entire subject of integration.

Arc length, partial circumferences

An arc is a piece of the boundary subtended by a central angle $\theta$. If $\theta$ is measured in degrees, the arc occupies the fraction $\theta/360$ of the full revolution:

$\ell = \frac{\theta}{360} \cdot 2\pi r.$

In radians (which we will introduce properly in §11), the formula simplifies to $\ell = r\theta$, one of the cleanest formulas in mathematics, and the reason radians are preferred over degrees once you start doing calculus.

Try it

• Predict first: what is the circumference of a unit circle? Set $r = 1$ and verify the readout shows $2\pi \approx 6.28$.
• Before adjusting: if you triple $r$, by what factor should the circumference change? Set $r = 3$ and verify it reads $6\pi \approx 18.85$, tripled, because circumference is linear in $r$.
• A wheel has diameter $2$ m. One full rotation moves it $2\pi \approx 6.28$ m forward, since the wheel rolls along a length equal to its circumference.

A trap to watch for

If you only remember one circle formula, you will sooner or later use the wrong one. Common slips: writing $\pi r^2$ for the perimeter of a track, or $2\pi r$ for the area of a pizza. Quick mnemonic: circumference is one-dimensional (it has units of length), so it is linear in $r$; area is two-dimensional (square units), so it is quadratic in $r$. When in doubt, check the units of your answer.

What you now know

You can compute the circumference of any circle and the arc length of any sector, you understand the calculus-flavoured connection $dA/dr = C$, and you have a units-based sanity check to keep area and circumference apart. The next chapter places the circle on a coordinate plane and rewrites everything algebraically.

Quick check

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 7, §2, circumference and the $dA/dr = C$ derivation.
• Archimedes (c. 250 BCE). Measurement of a Circle, Proposition 1 (Heath translation, Dover, 1953). The first rigorous bound on $\pi$.
• Stewart, J. (2015). Calculus, 8th ed. Cengage. Chapter 1 introduces arc length and the connection to integration.
• Spivak, M. (2008). Calculus, 4th ed. Publish or Perish. Appendix on "Pi" gives a complete modern construction of $\pi$ as the period of $\sin$.