Circle Calculator: Chord Length from Radius & Angle

This circle calculator finds the length of a chord when you enter the circle’s radius and either the central angle or the distance from the center to the chord. You get the chord length in the same unit you used for the radius, plus the arc length and sector area for that angle. It’s a straightforward geometry tool for anyone working with circles in construction, design, or math homework.

Formula

The chord length \(c\) is given by:

\(c = 2r \sin\left(\frac{\theta}{2}\right)\)

where:

• \(r\) = radius of the circle (any unit, e.g., meters, feet, inches)
• \(\theta\) = central angle subtended by the chord (in degrees)
• \(c\) = chord length (same unit as radius)

This formula comes from splitting the isosceles triangle formed by two radii and the chord into two right triangles. The sine of half the angle gives half the chord, so you double it.

Worked Example

Using the default values: radius \(r = 5\) m, central angle \(\theta = 60^\circ\).

1. Half the angle: \(\theta/2 = 30^\circ\)
2. \(\sin(30^\circ) = 0.5\)
3. \(2 \times 5 \times 0.5 = 5\) m

The chord length is 5.00 m. For a 60° angle in a circle of radius 5 m, the chord is exactly equal to the radius (this happens only at 60°).

When to Use This Calculator

Use this when you know the radius and the central angle (the angle between two radii that meet the chord’s endpoints). Common situations:

• Laying out an arch or curved wall with a given span angle
• Finding the width of a circular segment (e.g., a flat cut across a pipe)
• Calculating the distance between two points on a circle’s circumference
• Designing gears, pulleys, or decorative circular patterns

If instead you know the distance from the center to the chord (the sagitta’s complement), you’d use a different formula: \(c = 2\sqrt{r^2 – d^2}\), where \(d\) is the perpendicular distance from center to chord. This calculator uses the angle version, which is more common when you’re working with angular measurements.

How to Read the Result

The chord length is a straight-line distance between two points on the circle’s edge. It’s always less than or equal to the diameter (2r). For a 180° angle, the chord equals the diameter. For very small angles, the chord is nearly equal to the arc length, but never exactly (except at 0°).

The arc length is the distance along the curved edge, and the sector area is the pie-slice region between the two radii and the arc. These come in handy when you’re comparing straight vs. curved paths or calculating material for a circular segment.

Common Mistakes and Limitations

• Using degrees vs. radians: The formula above uses degrees. If you have radians, multiply by \(180/\pi\) or use the radian version \(c = 2r\sin(\theta/2)\) with \(\theta\) in radians.
• Confusing chord with arc: The chord is the straight line; the arc is the curved line. For a 60° angle in a 5 m radius, the chord is 5 m, but the arc is \(5\pi/3 \approx 5.24\) m.
• Unit mix-ups: If you enter radius in meters and angle in degrees, the chord comes out in meters. Don’t mix feet and inches without converting.
• Rounding too early: Keep extra digits in intermediate steps, especially for small angles, to avoid significant error in the final result.

The formula assumes a perfect circle. Real-world objects like pipes or arches may have slight deformations, but for most practical purposes, the error is negligible.

Alternative Approaches

If you don’t have the central angle, you can measure the chord directly with a tape measure. Or, if you know the sagitta (the maximum height of the arc above the chord), you can compute the chord using:

\(c = 2\sqrt{2rs – s^2}\)

where \(s\) is the sagitta. This is useful in woodworking or when measuring an existing arch. Another method is using the formula \(c = 2r\sin(\arccos(d/r))\) if you know the distance \(d\) from center to chord.

Frequently Asked Questions

What is a chord in a circle?

A chord is a straight line segment whose endpoints both lie on the circle’s circumference. The longest chord is the diameter, which passes through the center.

How do I find the chord length without the angle?

If you know the radius and the perpendicular distance from the center to the chord, use \(c = 2\sqrt{r^2 – d^2}\). If you know the sagitta (height of the arc), use \(c = 2\sqrt{2rs – s^2}\).

What is the relationship between chord length and arc length?

The chord is always shorter than the arc for any angle between 0° and 360°. The arc length is proportional to the angle (arc = \(r\theta\) in radians), while the chord grows with the sine of half the angle. For small angles, they are close; for 180°, the chord equals the diameter and the arc is half the circumference.

Can the chord be longer than the radius?

Yes. The chord can be any length from 0 (for a 0° angle) up to the diameter (2r) for a 180° angle. It equals the radius only when the central angle is exactly 60°.

Does this work for any unit?

Yes, as long as you use the same unit for radius and chord. The angle is dimensionless (degrees), so you can mix metric and imperial for the radius and get the chord in that same unit.

Why is the formula \(2r\sin(\theta/2)\)?

Because two radii and the chord form an isosceles triangle. Dropping a perpendicular from the center to the chord bisects the angle and the chord, creating two right triangles. In each, the opposite side is half the chord, and the hypotenuse is the radius, so \(\sin(\theta/2) = (c/2)/r\). Solving for \(c\) gives \(2r\sin(\theta/2)\).

Related

• Area Ratio Calculator: How to Use It
• Area Calculator Square: Formula & Examples
• Area Calculator: How to Measure Any Shape

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 | (pi * r * a) / 180 | 2
out sector | Sector area | m² | (pi * r^2 * a) / 360 | 2