How do I use the Arc Length Calculator?

Input any two of the following parameters: central angle (θ), radius (r), or chord length (c). The calculator will compute the remaining values and display them visually and numerically.

What formulas are used in this calculator?

Key formulas include: Arc Length: L = r × θ (θ in radians); Chord Length: c = 2 × r × sin(θ/2); Segment Area: S = (1/2) × r² × (θ – sin(θ)); Sagitta (height from chord to arc): h = r × (1 – cos(θ/2)).

The sagitta is the perpendicular distance from the midpoint of the chord to the arc. It represents the height of the arc above the chord and is crucial in applications like arch design.

How does changing the radius or angle affect the arc?

Increasing the radius or central angle lengthens the arc and brings the chord closer to the diameter. Keeping the chord constant and shrinking the radius increases the central angle.

Arc Length Calculator with Animation

Q: What are the practical applications of this calculator?

This tool is useful for designing arches, circular openings, greenhouse frames, gears, and other curved elements where precise geometry is required.

Angle:

Radius:

Chord:

Angle
Radius
Chord
Arc Length
Sector Area
Height from Chord

This interactive tool draws the geometry of a circular arc and calculates its main measures from any two of these inputs: angle, radius or chord length. Use it for designing arches, circular openings, greenhouse frames, gears and other curved elements where precise geometry is required. The controls let you change angle, radius or chord. All dependent values update immediately and the diagram refreshes to match. Results are shown numerically and visually.

Table of Contents

What you can compute

Arc length from radius and central angle.
Chord length from radius and angle.
Central angle from chord and radius.
Area of the circular segment under the arc.
Rise of the arc above the chord (sagitta).

Key formulas

Arc length
$$L = r \,\theta$$

Chord length
$$c = 2\,r\,\sin\!\left(\frac{\theta}{2}\right)$$

Segment area
$$S = \frac{1}{2}\,r^{2}\,\bigl(\theta – \sin\theta\bigr)$$

Sagitta (height from chord to arc)
$$h = r\Bigl(1 – \cos\!\frac{\theta}{2}\Bigr)$$

All formulas use the central angle $ \theta $ in radians. Convert degrees $ \alpha $ to radians by

$$\theta = \frac{\pi\,\alpha}{180}$$

Example

Take radius r = 7 m and central angle 60° (that is $ \theta = \pi/3 $ radians).

Arc length
$$L = r\theta = 7 \cdot \frac{\pi}{3} \approx 7.33 m$$

Chord
$$c = 2\cdot7\cdot\sin(\pi/6) = 7.00 m$$

Segment area
$$S = \frac{1}{2}\cdot49\cdot\bigl(\frac{\pi}{3} – \sin\!\frac{\pi}{3}\bigr) \approx 4.44 m²$$

Sagitta
$$h = 7\bigl(1 – \cos(\pi/6)\bigr) \approx 0.938 m$$

Parameter relations

Parameter	Relation	Dependence
Arc length	$L = r\theta$	Proportional to radius and angle
Chord length	$c = 2r\sin(\theta/2)$	Grows with angle
Segment area	$S = \frac{1}{2} r^2(\theta – \sin\theta)$	Increases as angle increases
Sagitta	$h = r(1 – \cos(\theta/2))$	Approaches r as angle nears 180°

For fixed radius, increasing the angle lengthens the arc and brings the chord closer to the diameter.
Keeping the chord constant and shrinking the radius increases the central angle.
Sagitta is critical when fitting curved members into frames; small changes in angle or radius may noticeably change the fit.