Visual walkthrough — Arc length and sector area using radians

We only assume you can draw a circle and measure a length with a piece of string. Everything else — the word angle, the symbol $\theta$, the number $\pi$ — gets built in front of you.

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Step 1 — What is a circle, and what is the radius?

WHAT: Draw a dot (the centre) and sweep a fixed-length stick around it. The tip traces a circle. The length of that stick is the radius, written $r$.

WHY: Every formula on this page is measured in "sticks" — copies of the radius. So the radius is our ruler. Before we can measure arcs or areas in terms of $r$, we must fix what $r$ is.

PICTURE: the stick $r$ points from centre $O$ out to the edge. Wherever the tip lands, the distance back to $O$ is always the same $r$.

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Step 2 — What is an angle, and why measure it with the arc?

WHAT: Open two radii from the centre like a pair of scissors. The amount of "opening" between them is the angle, which we name $\theta$ (the Greek letter theta, just a label for "the angle"). The curved edge trapped between the two radii is the arc, with length $s$.

WHY THIS WAY of measuring: A degree is an arbitrary human choice — someone once decided a full turn is $360$ pieces. But the circle itself offers a natural ruler: how many radius-lengths of arc did the opening carve out? That count is the angle in radians. Nothing arbitrary — the shape measures itself.

PICTURE: the blue arc $s$ sits on the curved edge; the two pale-yellow radii are its straight sides. $\theta$ is the wedge between them.

Link back: this is exactly the radian idea the parent used.

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Step 3 — Why the full turn is $2\pi$ radians

WHAT: Keep opening the scissors until they come all the way back around — a full turn. The arc is now the entire circumference (the whole rim). Its length is $2\pi r$.

WHY $2\pi r$ and where $\pi$ comes from: $\pi$ is defined as "how many diameters fit around the rim." The diameter is $2r$, and $\pi$ of them make the circumference, so rim $=\pi \times 2r=2\pi r$. (This is the circumference fact — we borrow it, we don't re-derive it here.)

WHY it matters for us: Using Step 2's ruler, the full-turn angle counts how many radius-lengths fit around the rim: ${\theta }_{\text{full}}=\frac{\text{whole rim}}{r}=\frac{2\pi r}{r}=2\pi .$ The $r$ cancels — a full turn is $2\pi$ radians, no matter how big the circle.

PICTURE: the rim wrapped by $2\pi \approx 6.28$ copies of the radius laid end to end.

Recall Why doesn't the circle's size change the answer?

Because we measure arc in units of the radius itself. A bigger circle has a longer rim, but its radius-ruler is longer too, in exactly the same proportion. The count $2\pi$ stays fixed. ::: The radius appears in both the arc and the ruler, so it cancels.

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Step 4 — The proportion that gives arc length $s=r\theta$

WHAT: A partial opening $\theta$ carves an arc that is a fraction of the whole rim. That fraction is (your angle) out of (a full turn): $\frac{\stackrel{\text{your arc}}{\overbrace{s}}}{\underset{\text{whole rim}}{\underbrace{2\pi r}}}=\frac{\stackrel{\text{your angle}}{\overbrace{\theta }}}{\underset{\text{full turn}}{\underbrace{2\pi }}}.$

WHY a proportion and not something fancier: The rim is uniform — no part of it is more curved than another. So "one third of the angle" traces "one third of the rim." Equal fractions on both sides. That single fact is the whole engine.

Now solve for $s$ — multiply both sides by $2\pi r$: $s=2\pi r\cdot \frac{\theta }{2\pi }=r\theta .$ Term by term: the $2\pi$ in the numerator (from the rim) cancels the $2\pi$ in the denominator (from the full turn). What survives is one lonely $r$ multiplying $\theta$.

PICTURE: two pie slices — a quarter-turn arc is a quarter of the rim; a half-turn arc is half the rim. Fraction of angle $=$ fraction of arc.

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Step 5 — The same proportion gives sector area $A=\frac{1}{2}{r}^2\theta$

WHAT: The sector is the filled pizza-slice between the two radii. It is the same fraction of the whole disc that the arc was of the whole rim: $\frac{\stackrel{\text{your slice}}{\overbrace{A}}}{\underset{\text{whole disc}}{\underbrace{\pi r^2}}}=\frac{\theta }{2\pi }.$

WHY the disc area is $\pi r^2$: we borrow it from the area fact. The key new character is the $\pi r^2$ on the left instead of $2\pi r$.

Now solve for $A$ — multiply both sides by $\pi r^2$: $A=\pi r^2\cdot \frac{\theta }{2\pi }=\frac{\pi }{2\pi }{r}^2\theta =\frac{1}{2}{r}^2\theta .$ Term by term: this time only one $\pi$ cancels (top vs bottom), and the $2$ underneath has no partner — so it lands in front as a $\frac{1}{2}$. The radius, meanwhile, arrived squared from the disc area and stays squared.

PICTURE: unroll the sector into a near-triangle of base $s=r\theta$ and height $r$. A triangle's area is $\frac{1}{2}(\text{base})(\text{height})=\frac{1}{2}(r\theta )(r)=\frac{1}{2}{r}^2\theta$ — the exact same answer, seen geometrically.

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Step 6 — Degenerate and edge cases (never let the reader fall off)

WHAT / WHY / PICTURE — every extreme checked:

• $\theta =0$ (scissors shut): $s=r\cdot 0=0$, $A=\frac{1}{2}{r}^2\cdot 0=0$. No opening, no arc, no slice. ✔
• $\theta =2\pi$ (full turn): $s=r\cdot 2\pi =2\pi r$ (the whole rim, matches Step 3), $A=\frac{1}{2}{r}^2(2\pi )=\pi r^2$ (the whole disc). ✔ The formulas close the loop onto the full-circle facts.
• $\theta >2\pi$ (over-wound): the formula keeps giving bigger numbers — you've simply wrapped past the start. Fine for total distance travelled, but for a drawn region, reduce $\theta$ below $2\pi$ first.
• Tiny $\theta$: the arc becomes almost a straight chord — this is the doorway to $s \approx r\theta$ used as a length.
• Angle in degrees: the proportion in Steps 4–5 used "$2\pi =$ full turn." If your $\theta$ is in degrees, the full turn is $360$, not $2\pi$, so the cancellation breaks. Convert first: ${\theta }_{\text{rad}}={\theta }_{\text{deg}}\times \frac{\pi }{180}$.

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The one-picture summary

Everything on one board: the radius-ruler (Step 1–2) sets the scale; the same proportion (Steps 4–5) sends the arc to $r\theta$ and the sector to $\frac{1}{2}{r}^2\theta$; the edge cases (Step 6) snap onto the full-circle facts.

Recall Feynman retelling — the whole walkthrough in plain words

I draw a dot and swing a stick of length $r$ around it — that's my circle, and the stick is my ruler. When I open two sticks like scissors, the curved edge between them is the arc. I measure the opening not in made-up degrees but in "how many sticks of arc" it carves — that count is the angle $\theta$ in radians. Going all the way around uses $2\pi$ sticks, because the rim is $2\pi r$ long and each stick is $r$. Any partial opening is just a fraction of that full turn, and — because the rim is perfectly even — it traces the same fraction of the rim. Solve that proportion and one $r$ survives: $s=r\theta$. Do the identical fraction trick on the filled slice against the whole disc $\pi r^2$; one $\pi$ cancels, a leftover $2$ becomes a $\frac{1}{2}$, and the radius comes squared because area spreads in two directions. That's $A=\frac{1}{2}{r}^2\theta$. Check the ends: shut the scissors and everything is zero; open them fully and I recover the whole rim and the whole disc. The formulas were the circle describing itself the entire time.

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Connections

• Arc length and sector area using radians (parent)
• Radian measure and degree conversion
• Circumference and area of a circle
• Small-angle approximations
• Angular velocity
• Segment area