Arc length of a circle: L = r × θ (radians)

Radian formula

$L = r \times \theta$

• $r$: radius
• $\theta$: angle in radians
• $L$: arc length

Degree formula

$L = \dfrac{\alpha}{360} \times 2 \pi r$

• $\alpha$: angle in degrees
• Fraction of the full perimeter ($2\pi r$).

Reminder: degrees ↔ radians

$\pi \text{ rad} = 180°$

• $\theta_{rad} = \alpha_{deg} \times \pi / 180$
• $\alpha_{deg} = \theta_{rad} \times 180 / \pi$

Common angles:

Degrees	Radians
30°	π/6
45°	π/4
60°	π/3
90°	π/2
180°	π
360°	2π

Examples

Arc with radius 5 cm, angle 60°:

• In radians: $60° = \pi/3$, $L = 5 \times \pi/3 \approx 5.24$ cm.
• In degrees: $L = (60/360) \times 2\pi \times 5 = (1/6) \times 10\pi \approx 5.24$ cm. ✓

Full perimeter ($\alpha = 360°$, $\theta = 2\pi$): $L = r \times 2\pi = 2\pi r$. ✓

Area of a circular sector

$\mathcal{A} = \dfrac{r^2 \theta}{2}$

(radians). Or $\mathcal{A} = \dfrac{\alpha}{360} \times \pi r^2$ (degrees).

Common mistakes

1. Using $L = r \theta$ with $\theta$ in degrees (wrong: convert to radians).
2. Forgetting the radius in the formula.
3. Confusing arc length with sector area.

?À toi de jouer

Arc with radius 10 cm, angle 90°: length?

A.5π cmB.10π cmC.20π cmD.π cm

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