Chord Length Calculator: Formula & Examples

This chord length calculator works out the straight-line distance between two points on a circle’s circumference. You enter the circle’s radius and the central angle (the angle between the two radius lines that meet those points), and you get back the chord length in the same unit you used for the radius. The calculation uses the standard geometric formula c = 2r × sin(θ/2).

Formula

The chord length is found with the equation:

c = 2 × r × sin(θ / 2)

Where:

• c = chord length (same unit as the radius, e.g., meters, feet, inches)
• r = radius of the circle (any unit of length)
• θ = central angle in degrees (the angle at the center of the circle between the two radii that touch the chord’s endpoints)

If your angle is in radians, use c = 2r × sin(θ/2) directly with θ in radians. The calculator above expects degrees and converts internally using the sine function for degrees.

Worked Example

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

1. Divide the angle by 2: 60° / 2 = 30°
2. Find the sine of 30°: sin(30°) = 0.5
3. Multiply by 2 and by the radius: 2 × 5 × 0.5 = 5

The chord length is 5.00 m. This matches the calculator’s output. Notice that for a 60° angle in a circle of radius 5, the chord equals the radius—this happens only because sin(30°) = 0.5, making the chord exactly r.

When to Use a Chord Length Calculator

Chord lengths appear in many practical fields:

• Construction and woodworking: finding the straight distance between two points on an arch or circular frame.
• Engineering: calculating distances between bolt holes on a flange or the length of a structural member across a curved surface.
• Design: laying out circular patterns, gears, or decorative arcs.
• Astronomy and navigation: estimating the distance across a circular orbit or the visible chord of a planet’s disk.
• Mathematics education: solving geometry problems involving circles, inscribed angles, or sectors.

In short, any time you have a circle and need the straight line connecting two points on its edge, you need the chord length.

How to Read the Result

The output is a length, so it uses the same unit you entered for the radius. If you entered meters, the chord is in meters; if you entered feet, it’s in feet. The chord length is always less than or equal to the diameter (2r). It equals the diameter only when the central angle is 180° (a straight line through the center). As the angle approaches 0°, the chord length approaches 0. The sagitta (second output) is the maximum distance from the chord to the arc—useful when you need the height of a circular segment.

Common Mistakes and How to Avoid Them

• Mixing degrees and radians: The formula works with either, but you must be consistent. If your calculator is in radian mode, convert degrees to radians first (multiply by Ï€/180). The calculator above handles degrees automatically.
• Using the diameter instead of the radius: Double-check which one you have. If you only have the diameter, divide it by 2 to get the radius.
• Forgetting to halve the angle: The formula uses half the central angle, not the full angle. This is a common error when using a scientific calculator.
• Rounding too early: If you round sin(30°) to 0.3, you’ll get a wrong answer. Keep full precision until the final step.

Limitations of the Formula

The chord length formula assumes a perfect circle. Real-world objects like pipes, arches, or planetary orbits may deviate slightly from a true circle due to manufacturing tolerances, deformation, or elliptical shapes. For an ellipse, the chord length is not given by this formula; you would need a different approach (often numerical integration). Also, the central angle must be between 0° and 360°. Angles greater than 360° wrap around, but the chord for an angle of 400° is the same as for 40°, since the sine function repeats every 360°.

Frequently Asked Questions

What is the difference between a chord and an arc?

An arc is the curved portion of the circle’s circumference between two points. A chord is the straight line segment that connects those two points. The arc is longer than the chord (unless the points are identical). The chord is always inside the circle, while the arc lies on the circumference.

Can I calculate chord length if I only know the arc length?

Yes, if you also know the radius. The arc length (s) is related to the central angle by s = r × θ (θ in radians). Solve for θ = s/r, then plug into the chord formula. If you don’t have the radius, you cannot determine the chord length from arc length alone.

How do I find the chord length without a calculator?

You can use a trigonometric table or a rough mental estimate. For common angles like 30°, 45°, 60°, and 90°, the sine values are well-known. For example, sin(30°)=0.5, sin(45°)≈0.7071, sin(60°)≈0.8660. Then apply c = 2r × sin(θ/2). For other angles, you’ll need a calculator or a table.

What is the sagitta and why is it useful?

The sagitta is the perpendicular distance from the midpoint of the chord to the arc. It’s also called the arc height. It’s useful in construction and optics when you need to know how much a curved surface rises above its base. The formula is h = r – r × cos(θ/2), which is the second output in the calculator.

Does the chord length formula work for any circle size?

Yes, it works for any circle, from a tiny gear to a planetary orbit, as long as you use consistent units. The formula is scale-invariant: if you double the radius, the chord length doubles for the same angle.

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in r | Radius | m | 5 | 0
in a | Central angle | degrees | 60 | 0
out c | Chord length | m | 2 * r * sind(a / 2) | 2
out h | Sagitta (arc height) | m | r * (1 - cosd(a / 2)) | 2