Free Chord Length Calculator

Chord Length Explained – Formulas, Examples, and Online Calculator

This free chord length calculator provides a quick, reliable way to compute the circle chord length using your preferred input – the sagitta (chord height), the apothem, or the central angle. Alongside the chord length, the tool reveals other essential circle metrics: the diameter, circumference, total area, arc length, sector area, segment area, and the sagitta itself. Whether you are a student tackling geometry homework, an engineer designing curved structures, or a DIY enthusiast working with arches, this online sagitta calculator streamlines the computation.

Basic Definitions

Within a circle, a chord is a straight segment whose endpoints lie on the circumference. Unlike the diameter, it does not need to cross the center. The perpendicular distance from the chord to the circle’s center is the apothem ($a$). The perpendicular distance from the chord to the arc directly above it is the sagitta or chord height ($h$). The central angle ($\alpha$) subtended by the chord is the angle formed by the two radii at the center that connect to the chord’s endpoints. For a chord that creates a minor arc (central angle less than 180°), the radius can be expressed as $r = a + h$. This relationship is helpful when converting between apothem and sagitta.

The Three Main Chord‑Length Formulas

All formulas require the radius ($r$). Once the radius is known, use the appropriate expression:

Input Parameter	Chord Length Formula
Sagitta $h$ (chord height)	$c = 2\sqrt{2rh - h^{2}}$
Apothem $a$	$c = 2\sqrt{r^{2} - a^{2}}$
Central angle $\alpha$ (radians)	$c = 2r \sin\!\left(\dfrac{\alpha}{2}\right)$

If the central angle is given in degrees, multiply by $\pi/180$ to obtain radians.

Deriving the Radius from Other Quantities

When the radius is not directly known, it can be computed from the following circle measurements:

Known Parameters	Radius Expression
Area $A$	$r = \sqrt{A / \pi}$
Circumference $C$	$r = C / (2\pi)$
Diameter $d$	$r = d / 2$
Arc length $L$ and central angle $\alpha$ (rad)	$r = L / \alpha$
Sector area $A_s$ and central angle $\alpha$	$r = \sqrt{2A_s / \alpha}$
Segment area $A_m$ and central angle $\alpha$	$r = \sqrt{2A_m / (\alpha - \sin\alpha)}$

After obtaining the radius, you can plug it into any of the three chord formulas above.

Quick Reference: Chord Length Factors for Common Angles

The multiplier below (equal to $2\sin(\alpha/2)$) gives the chord length when multiplied by the radius.

Central Angle (°)	Chord Length Factor
30°	0.5176
45°	0.7654
60°	1.0000
90°	1.4142
120°	1.7321
180°	2.0000

Example: A circle with radius 10 and a 90° central angle produces a chord length of $10 \times 1.4142 = 14.14$ units.

Worked Examples

Example 1 – Using the central angle:
Radius = 8 m, central angle = 45°. Convert 45° to radians ($\pi/4$) and compute:

$$c = 2 \times 8 \times \sin(22.5^\circ) \approx 16 \times 0.3827 = 6.12\ \text{m}.$$

Example 2 – Using the sagitta:
Radius = 10 cm, sagitta = 3 cm.

$$c = 2\sqrt{2 \times 10 \times 3 - 3^{2}} = 2\sqrt{60 - 9} = 2\sqrt{51} \approx 14.28\ \text{cm}.$$

Example 3 – Using the apothem:
Radius = 6 in, apothem = 4 in.

$$c = 2\sqrt{6^{2} - 4^{2}} = 2\sqrt{36 - 16} = 2\sqrt{20} \approx 8.94\ \text{in}.$$

These examples demonstrate how the calculator handles inputs of any type and delivers exact chord lengths.

How to Use the Free Online Chord Length Calculator

1. Enter the radius directly in the designated field. If the radius is unknown, activate the “Other parameters” box and supply one of: area, circumference, diameter, arc length, sector area, or segment area. The tool automatically calculates the radius.
2. Provide one of the three trigger parameters: chord height (sagitta), apothem, or central angle.
3. Obtain the result – the chord length is displayed as soon as the required fields are filled.

Additional Parameters Automatically Computed

With the radius and one trigger parameter known, the calculator also shows:

• Diameter ($2r$)
• Circumference ($2\pi r$)
• Total area ($\pi r^{2}$)
• Arc length ($r\alpha$)
• Sector area ($\tfrac{1}{2}r^{2}\alpha$)
• Segment area ($\tfrac{1}{2}r^{2}(\alpha - \sin\alpha)$)
• Sagitta – the chord height itself.

Special Case: The Diameter as a Chord

When the chord passes through the circle’s center, it becomes a diameter. In this situation:

• The chord length is at its maximum: $2r$.
• The sagitta equals $r$.
• The apothem is zero.
• The central angle is $180°$ ($\pi$ rad).

This limit provides a quick sanity check for both manual calculations and the calculator’s output.

Why “Sagitta”?

The name sagitta comes from Latin for “arrow”. Imagine a bow: the taut string is the chord, the bow’s curve is the arc, and the arrow stands upright at the midpoint, connecting the string with the bow. That vertical distance is precisely the chord height. This historical analogy makes the concept easy to visualize.

Selecting the Appropriate Formula

Your choice depends entirely on the data available:

• Sagitta formula – use when you can measure the rise of the arc above the chord (e.g., in arch construction).
• Apothem formula – use when the perpendicular distance from the center to the chord is accessible (common in symmetric designs).
• Central‑angle formula – use when the angle is given in blueprints or when the chord’s endpoints are located at known angular positions.

All three methods are equivalent, and this circle chord length and sagitta calculator supports them all with equal ease.

Whether you need a precise value for a mathematical proof, a building plan, or a creative project, this free chord length calculator delivers accurate results instantly and eliminates time‑consuming manual algebra.