Circle Circumference: C = 2πr

The circumference of a circle is famously $C = 2\pi r$ (or equivalently $C = \pi d$, where $d$ is the diameter). The constant $\pi \approx 3.14159$ shows up because every circle has the same ratio of circumference to diameter — that ratio defines $\pi$.

The Definition of π

For any circle:

$\pi = \dfrac{C}{d} = \dfrac{C}{2r}$

So $C = 2\pi r$. This isn't a derivation — it's literally the definition of $\pi$. Geometrically, $\pi$ is "how many diameters' worth of arc you wrap around to get back where you started."

Step-by-Step Solution

Given: A circle of radius $r = 1$.

Find: The circumference, and verify by approximating with inscribed polygons.

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Step 1 — Apply the formula.

$C = 2\pi r = 2\pi(1) = 2\pi \approx 6.2832$

Step 2 — Approximate by inscribing regular polygons.

The circumference of a regular $n$-sided polygon inscribed in a unit circle is:

$C_n = 2n\sin\!\left(\dfrac{\pi}{n}\right)$

(Each side is the chord of an angle $2\pi/n$; using the chord length formula gives $2\sin(\pi/n)$.)

Tabulate for several $n$:

• $n = 4$ (square): $C_4 = 8\sin(\pi/4) \approx 5.657$
• $n = 6$ (hexagon): $C_6 = 12\sin(\pi/6) = 6.000$
• $n = 12$: $C_{12} = 24\sin(\pi/12) \approx 6.211$
• $n = 24$: $C_{24} \approx 6.265$
• $n = 48$: $C_{48} \approx 6.279$
• $n = 100$: $C_{100} \approx 6.282$
• $n \to \infty$: $C_n \to 2\pi \approx 6.283$

This is exactly how Archimedes computed $\pi$ around 250 BC, eventually getting it to $3.1408 < \pi < 3.1429$ using a 96-sided polygon.

Step 3 — Verify the approximation rate.

The polygon underestimates the circle (chord length is less than arc length). The error shrinks as $1/n^{2}$, so each doubling of $n$ gives 4× the accuracy.

Step 4 — Compute for some real-world examples.

• A bicycle wheel of radius $r = 0.35\;\text{m}$: $C = 2\pi(0.35) \approx 2.199\;\text{m}$ — that's how far you go per revolution.
• The Earth's equator (radius $\approx 6378\;\text{km}$): $C \approx 40{,}075\;\text{km}$.
• An atomic nucleus (radius $\sim 10^{-15}\;\text{m}$): $C \sim 6 \times 10^{-15}\;\text{m}$.

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Answer:

$\boxed{\;C = 2\pi r = 2\pi \approx 6.2832\;}$

For any circle, the circumference is exactly $2\pi$ times the radius — that ratio is what $\pi$ means. As you inscribe polygons with more and more sides, their perimeters converge to this value.

Try It

• Adjust the number of polygon sides to see how quickly the polygon perimeter approaches the true circumference.
• Watch the polygon "round out" into a circle as $n$ grows.
• At $n = 100$, the polygon is visually indistinguishable from the circle.