Arc Length to Straight Line Calculator

This calculator converts the length of an arc (a curved path along a circle) into the straight-line distance between its two endpoints, commonly called the chord length. You enter the radius of the circle and either the arc length or the central angle, and you get the straight-line distance, along with the angle in degrees if needed.

Formula

The relationship between arc length and the straight line connecting its endpoints comes from the geometry of a circle. If you know the radius r and the central angle θ (in radians), the arc length is:

Arc length = r × θ

The straight-line distance between the endpoints is the chord length, given by:

Chord length = 2 × r × sin(θ / 2)

Combining these, if you know the arc length a and radius r, the chord length is:

Chord = 2 × r × sin(a / (2 × r))

Here, r is the radius (any length unit), a is the arc length (same unit as radius), and the chord length comes out in the same unit. The angle must be in radians for the sine function; the calculator converts arc length to angle automatically using θ = a / r.

Worked Example

Using the default values: radius = 5 m, arc length = 7.85 m.

1. Find the central angle: θ = a / r = 7.85 / 5 = 1.57 radians.
2. Convert to degrees if you like: 1.57 × 180 / π ≈ 90°.
3. Compute half the angle: 1.57 / 2 = 0.785 radians.
4. Sine of 0.785 ≈ 0.7071.
5. Chord length = 2 × 5 × 0.7071 = 7.071 m.

The calculator shows 7.07 m for the chord and 90.0° for the angle, matching this step-by-step result.

When to Use This Calculator

You need this conversion whenever you have a curved path and want to know the direct “as-the-crow-flies� distance between its ends. Common situations include:

• Measuring a road or trail curve on a map and wanting the straight distance between two points.
• Calculating the width of a circular segment (e.g., the flat width of a curved piece of material).
• Determining the cable length needed to span a curved conduit or pipe.
• Checking the chord length for a circular arch or curved window.

The calculator works for any circle, whether it’s a full circle, a semicircle, or a tiny arc. Just make sure the arc length is less than the full circumference (2πr) for a valid single arc.

How to Choose Your Inputs

You only need two values: the radius and the arc length. The radius can be in any unit—meters, feet, inches, etc.—as long as the arc length uses the same unit. For example, if the radius is in feet, the arc length must also be in feet.

If you don’t know the arc length but you know the central angle, you can compute the arc length first using the formula a = r × θ (with θ in radians). Then enter that value here. The calculator does not accept angle input directly, but you can convert degrees to radians by multiplying by π/180.

For very small arcs (less than about 6°), the chord length is almost the same as the arc length. For larger arcs, the difference becomes significant. At 180° (a semicircle), the chord equals the diameter (2r), while the arc length is πr ≈ 3.14r.

Common Mistakes and Misconceptions

The most frequent error is mixing units. If you enter radius in meters and arc length in centimeters, the result will be wrong. Always convert to the same unit first.

Another common mix-up is using degrees in the sine function. The formula requires radians. This calculator handles that conversion internally, but if you try to do it by hand, remember that sin(90°) = 1, but sin(90 radians) is about 0.894. The calculator’s default example uses 1.57 radians, which is exactly π/2, so it works cleanly.

A third misconception is that the chord length is the same as the arc length. They are only equal when the arc is essentially a straight line (zero curvature). For any real curve, the chord is shorter than the arc, and the difference grows as the angle increases.

Finally, note that the chord length is the straight line between endpoints, not the “sagitta� (the height of the arc above the chord). If you need the sagitta, use the formula: sagitta = r − r × cos(θ/2).

Limitations of the Formula

This formula assumes a perfect circular arc. Real-world curves—roads, rivers, or irregular shapes—are rarely perfect circles, so the result is an approximation for those cases. For a road curve that is not a circular arc, you would need more advanced surveying methods.

The formula also fails if the arc length is greater than the circumference (2πr). In that case, the arc wraps around more than once, and the “straight line� concept becomes ambiguous. This calculator assumes a single arc of less than one full revolution.

For arcs larger than 180°, the chord length still works, but note that the chord is the shorter of the two possible straight lines between the endpoints (the other would go the long way around). The formula always gives the shorter chord.

Frequently Asked Questions

Q: Can I use this calculator for a semicircle?
Yes. For a semicircle, the central angle is 180° (π radians). If you enter the arc length as π × r (half the circumference), the calculator will return a chord of 2 × r, which is the diameter. For example, with r = 5 and arc length = 15.71, the chord is 10.00.

Q: What if I only know the central angle in degrees?
Convert degrees to radians by multiplying by π/180, then compute the arc length as a = r × θ. For instance, a 60° angle with r = 3 gives θ = 1.047 radians, arc length = 3.142, and chord = 3.00 (since 2 × 3 × sin(0.5236) = 3).

Q: Why is the chord length always shorter than the arc length?
Because the chord is a straight line, and the arc is a curved line connecting the same two points. A straight line is the shortest distance between two points in Euclidean geometry, so any curve between them is longer.

Q: Can I use this for a full circle?
No, because a full circle has no distinct endpoints—the start and end are the same point, so the chord length is zero. The formula would give zero if you entered the full circumference, but that’s not meaningful. Use this only for arcs less than 360°.

Q: What units should I use for the result?
The chord length comes out in the same unit you used for radius and arc length. If you enter meters, you get meters; if you enter feet, you get feet. The angle is always in degrees.

Q: Is there a way to find the radius if I only know the arc length and chord?
Yes, but it requires solving a transcendental equation: chord = 2r sin(a/(2r)). You can use numerical methods or trial and error. This calculator assumes you already know the radius, so you’d need to iterate manually.

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in r | Radius | m | 5 | 0.001
in a | Arc length | m | 7.85 | 0.001
out chord | Chord length | m | 2 * r * sin(a / (2 * r)) | 2
out angle | Central angle | deg | (a / r) * 180 / pi | 1