Circle Chord Length Calculator: Formula & Examples

This calculator finds the length of a chord in a circle when you enter the radius (or diameter) and either the central angle or the distance from the center to the chord. You get the chord length in the same unit you entered, plus the arc length and sector area for that angle. It uses the standard geometric formulas, so you can also use it to check your own hand calculations.

Formula

The chord length c is the straight-line distance between two points on a circle’s circumference. The most common formula uses the radius r and the central angle θ (in degrees):

c = 2 × r × sin(θ / 2)

where:

• c = chord length (same unit as r)
• r = radius of the circle (m, ft, cm, etc.)
• θ = central angle subtended by the chord, in degrees

If you know the diameter instead, simply use r = d/2. Another useful formula uses the perpendicular distance h from the circle’s center to the midpoint of the chord:

c = 2 × √(r² − h²)

This second version is handy when you can measure the sagitta (the height of the arc above the chord) or the distance from the center to the chord directly.

Worked Example

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

1. Convert the angle to half-angle: θ/2 = 30°.
2. Find sin(30°) = 0.5.
3. Multiply: c = 2 × 5 × 0.5 = 5 m.

The chord length is 5 m. The arc length is (60/360) × 2π × 5 ≈ 5.24 m, and the sector area is (60/360) × π × 25 ≈ 13.09 m². The calculator gives the same results (5.00 m, 5.24 m, 13.09 m²).

When to Use This Calculator

Chord length problems appear in woodworking, metalworking, architecture, and even in creating circular patterns or gears. You might need to find the distance between two holes on a flange, the width of a flat cut on a round bar, or the span of an arc in a bridge design. This calculator covers the two most common scenarios:

• You know the radius and the central angle (e.g., a 60° segment of a circle).
• You know the radius and the distance from the center to the chord (the perpendicular distance).

If you have the arc length and radius, you can still use the first formula by first calculating the angle: θ = (arc length / (2πr)) × 360°.

How to Read the Results

The chord length is always less than or equal to the diameter. When the angle is 180°, the chord equals the diameter (2r). When the angle is 0°, the chord is 0. The arc length is the distance along the curve, which is always longer than the chord for any angle between 0° and 360°. The sector area is the area of the pie-shaped slice enclosed by the two radii and the arc.

All results are in the same unit you entered for the radius. If you enter the radius in meters, the chord length and arc length are in meters, and the area is in square meters. For feet, the area is in square feet.

Common Mistakes and How to Avoid Them

• Mixing degrees and radians: The formula above uses degrees. If your calculator is in radian mode, you must convert: θ_rad = θ_deg × Ï€/180. This calculator uses degrees, so you don’t need to convert.
• Using diameter instead of radius: Always halve the diameter before using the formula. Forgetting this doubles your answer.
• Using the wrong formula: The chord length formula is not the same as the arc length formula. The chord is a straight line; the arc is curved.
• Rounding too early: If you round sin(θ/2) to two decimals before multiplying, you may lose precision. Keep full precision until the final step.

Limitations and Related Concepts

This calculator assumes a perfect circle and a chord that lies entirely inside it. For very small angles (less than about 1°), the chord length is nearly equal to the arc length, but they are never exactly the same. The formula works for any circle, regardless of size, because it is scale-invariant.

If you need to find the chord length from the sagitta (the height of the arc above the chord), you can use the second formula after solving for h: h = r − √(r² − (c/2)²). Conversely, if you have the sagitta s and radius r, the chord length is c = 2 × √(2rs − s²).

Frequently Asked Questions

How do I find chord length from the diameter and angle?

Divide the diameter by 2 to get the radius, then use the same formula: c = 2 × r × sin(θ/2). For example, if the diameter is 10 m and the angle is 90°, r = 5 m, so c = 2 × 5 × sin(45°) ≈ 7.07 m.

What is the difference between chord length and arc length?

A chord is the straight line connecting two points on the circle. The arc length is the distance along the curved circumference between those same two points. For any angle between 0° and 360°, the arc length is longer than the chord.

Can I use this for a semicircle?

Yes. For a semicircle, the central angle is 180°. The chord length becomes c = 2r × sin(90°) = 2r, which is exactly the diameter. The arc length is πr.

What if I only know the chord length and the radius? How do I find the angle?

Rearrange the formula: θ = 2 × arcsin(c / (2r)). For example, if c = 5 m and r = 5 m, then θ = 2 × arcsin(0.5) = 60°. Make sure your calculator is in degree mode.

Why does the chord length formula use sine?

The chord forms the base of an isosceles triangle with two radii as equal sides. Splitting that triangle in half creates a right triangle where the half-chord is the opposite side to the half-angle. The sine ratio gives the half-chord, so doubling it gives the full chord.

Is the chord length the same as the distance between two points on a circle?

Yes, exactly. The chord length is the straight-line distance between those two points. If you have two points on a circle and know the central angle between their radii, the chord length is the distance between them.

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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 area | Sector area | m^2 | (a / 360) * pi * r^2 | 2