Circle Arc Length Calculator
Calculate the curved arc length $s$ subtended by any central angle $\theta$, along with circular sector area, chord length, and circular segment area with exact $\pi$ multiples.
Interactive Circle Sector & Arc Canvas
Radius r & Subtended Arc sStep-by-Step Arc Length & Sector Geometry Derivations Formula: s = 2πr(θ/360°) = r⋅θ_rad
How to Calculate Circle Arc Length
To find the arc length of a circle segment, multiply the radius r by the central angle in radians: s = r·θ. If the angle is given in degrees, multiply the full circumference (2πr) by the angle's fraction of a complete circle (θ/360°): s = 2πr·(θ/360°) = (πrθ)/180°. For example, for a circle with radius 10 and central angle 90°, the arc length is s = 10·(π/2) = 5π ≈ 15.708 units.
Anatomy of Circle Arcs & Radians Definition
An arc is any portion of the circumference of a circle. The length of this curve depends directly on two geometric quantities: the size of the circle (measured by its radius $r$) and the opening of the angle at the center (the central angle $\theta$).
The concept of a radian is defined specifically so that $s = r\theta$: one radian is the central angle subtending an arc whose length equals the radius of the circle ($s = r$).
Linear distance along the curved edge.
2D surface area of the circular wedge.
Straight-line distance connecting arc tips.
Degree Formula vs Radian Formula (s = r⋅θ)
Fraction of full 360° circle times circumference.
Direct product of radius and radian measure.
Sector Area, Chord Length & Circular Segment
When analyzing a circle sector, three related geometric properties are frequently needed in civil engineering and architectural drafting:
- Sector Area: $A_{\text{sector}} = \frac{1}{2}r^2\theta = \pi r^2\left(\frac{\theta^\circ}{360^\circ}\right)$.
- Chord Length: $c = 2r\sin\left(\frac{\theta}{2}\right)$.
- Circular Segment Area: The region between the arc and chord: $A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = \frac{1}{2}r^2(\theta - \sin\theta)$.
Minor Arcs, Semicircles & Major Arcs
θ < 180° (π rad). The shorter arc connecting two points.
θ = 180° (π rad). Arc length s = πr.
θ > 180° (π rad). The longer arc (s = 2πr − s_minor).
Step-by-Step Worked Problems
Find the arc length of a circle of radius r = 6 cm with central angle θ = 60°.
1. Formula: s = 2πr × (θ / 360°).
2. Substitute: s = 2π(6) × (60 / 360) = 12π × (1/6) = 2π cm.
3. Numerical evaluation: s = 2(3.14159...) ≈ 6.283 cm.
Common Pitfalls: Angle Unit Conversion Errors
The short formula $s = r\theta$ is ONLY valid when $\theta$ is measured in radians. If $\theta$ is in degrees, plugging it into $s = r\theta$ produces an answer that is $\frac{180}{\pi} \approx 57.3$ times too large!
If a problem gives the diameter $d$, divide by 2 first ($r = d/2$) before computing arc length.
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