Chord of a Circle Calculator with Arc Length

This calculator finds the length of a chord in a circle when you enter the radius and either the central angle or the arc length. It also computes the arc length from the angle, or the angle from the arc length, so you can work in either direction. The result is given in the same unit you used for the radius (e.g., meters, feet, inches).

Formula

The chord length c for a circle of radius r and central angle θ (in degrees) is:

c = 2r sin(θ/2)

Here, r is the radius (any unit), θ is the central angle in degrees, and sin is the sine function evaluated in degrees. If you have the arc length s instead of the angle, first find the angle from θ = (360 × s) / (2πr), then use the same chord formula. The sagitta (the maximum distance from the arc to the chord) is r − r cos(θ/2).

Worked Example

Using the default values: radius = 5 m, central angle = 60°.

1. Half the angle: 60° / 2 = 30°.
2. sin(30°) = 0.5.
3. Chord length = 2 × 5 × 0.5 = 5 m.
4. Arc length = (60/360) × 2 × π × 5 = (1/6) × 31.4159 ≈ 5.24 m.
5. Sagitta = 5 − 5 × cos(30°) = 5 − 5 × 0.8660 ≈ 0.67 m.

The calculator shows chord = 5.00 m, arc = 5.24 m, sagitta = 0.67 m.

When to Use This Calculator

This tool is handy for construction, woodworking, metalworking, and any geometry problem where you need the straight-line distance across a circular segment. For example, you might need to cut a board to fit across a circular table, determine the width of a flat surface on a cylinder, or lay out an arch. It is also useful in engineering for calculating the chord length of a gear tooth or the width of a circular segment in a bridge truss.

You can enter either the central angle or the arc length. If you have the arc length, convert it to an angle first (as shown above), then use the chord formula. The calculator accepts any unit for the radius, as long as you use the same unit for the result.

How to Read the Results

The chord length is the straight line connecting the two endpoints of the arc. The arc length is the curved distance along the circle. The sagitta is the perpendicular distance from the midpoint of the chord to the arc (the “height� of the circular segment). In the example, a 60° arc on a 5 m radius circle has a chord of 5 m, an arc of 5.24 m, and a sagitta of 0.67 m.

Notice that the chord is always shorter than the arc (except when the angle is 0°, where both are 0). For a 180° angle (a semicircle), the chord equals the diameter (2r), and the arc length is πr. For a 360° angle, the chord is 0 because the two endpoints coincide, but the arc is the full circumference.

Common Mistakes and Unit Mix-ups

• Using radians instead of degrees: The formula here uses degrees. If you have the angle in radians, multiply by 180/Ï€ to convert to degrees, or use the radian version: c = 2r sin(θ/2) with θ in radians.
• Forgetting to halve the angle: The sine is applied to half the central angle, not the full angle. Using sin(60°) instead of sin(30°) gives 8.66 m instead of 5 m.
• Mixing units: If the radius is in feet and you want inches, convert first. The chord will be in the same unit as the radius.
• Arc length vs. chord length: They are different. The arc is the curved path; the chord is the straight line. Do not use one when the other is needed.

Limitations of the Formula

The formula assumes a perfect circle and a central angle between 0° and 360°. For angles greater than 360°, the chord repeats, but the arc length should be reduced modulo 360°. The sagitta formula works for angles up to 180°; for angles greater than 180°, the sagitta becomes larger than the radius, but the geometric interpretation changes (the chord is still defined, but the “height� is measured on the opposite side). In practice, most applications use angles less than 180°.

Also, the calculator gives exact mathematical results. Real-world measurements may include tolerances, material thickness, or stretching, so always verify with physical measurements for critical work.

Frequently Asked Questions

How do I find the chord length if I only have the arc length and radius?

First, convert the arc length to a central angle: θ = (360 × s) / (2πr). Then use the chord formula: c = 2r sin(θ/2). For example, if radius = 5 m and arc = 5.24 m, then θ ≈ 60°, and chord ≈ 5 m.

What is the difference between chord and arc length?

A chord is a straight line segment connecting two points on a circle. An arc is the curved portion of the circumference between those same two points. The arc is always longer than the chord for any non-zero angle.

Can I use this calculator for a semicircle?

Yes. Enter 180° as the central angle. The chord will be 2r (the diameter), and the arc length will be πr. For a radius of 5 m, the chord is 10 m and the arc is about 15.71 m.

What is the sagitta?

The sagitta is the perpendicular distance from the midpoint of the chord to the arc. It is also called the “height� of the circular segment. It is useful in optics and construction for determining the depth of a curved surface.

How accurate is the calculator?

The calculator uses standard mathematical functions and rounds to two decimal places. For most practical purposes, this is sufficient. If you need more precision, you can increase the decimal places in the output settings.

Can I enter the angle in radians?

This calculator expects degrees. If you have radians, multiply by 180/π to convert. For example, 1 radian ≈ 57.30°.

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in r | Radius | m | 5 | 0
in a | Central angle | degrees | 60 | 0
out chord | Chord length | m | 2 * r * sind(a / 2) | 2
out arc | Arc length | m | (a / 360) * 2 * pi * r | 2
out sag | Sagitta (height) | m | r - r * cosd(a / 2) | 2