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Chord Length Calculator

Find the length of a chord from the radius and central angle of a circle.

How it is calculated

The chord is the straight segment joining two points on a circle. The tool computes c = 2 × r × sin(θ / 2), where r is the radius (any length unit) and θ is the central angle in degrees, converted to radians internally. The answer is in the same unit as the radius and is shown with four decimals. The radius must be greater than zero, and the angle must be greater than 0 and at most 360; anything else returns no result. An angle of 180° gives the diameter (2r), the longest possible chord, and 360° gives 0 because both endpoints coincide. Angles above 180° give the same chord as 360° minus that angle.

Worked example

A circle has radius 10 and the chord spans a 90° central angle. Step 1: half the angle is 45°. Step 2: sin 45° ≈ 0.7071. Step 3: c = 2 × 10 × 0.7071 ≈ 14.14. With full precision the tool prints 14.1421, in the same unit as the radius.

Try this example →

Common mistakes

  • Entering the central angle in radians. The field expects degrees, so 1.57 would be read as a tiny 1.57° angle.
  • Entering the diameter as the radius, which doubles every result.
  • Using the inscribed angle (measured at the circumference) instead of the central angle. The inscribed angle is half of the central angle that subtends the same chord.
  • Forgetting that the angle is the full angle between the two radii, not the half-angle. The tool halves it for you.

About this calculator

The chord length formula, c = 2r × sin(θ/2), gives the straight-line distance between two points on a circle's circumference given the circle's radius r and the central angle θ that the chord subtends at the center — a direct consequence of splitting the isosceles triangle formed by the two radii and the chord into two right triangles.

Chord length calculations appear throughout engineering and design: civil engineers use them to lay out circular road curves and railway track curvature, mechanical engineers use them for gear tooth spacing and pulley design, and the same formula underlies CNC and CAD software when converting arc geometry into straight-line segments for machining or rendering.

Radius and central angle in degrees are all it takes: the chord, 2r × sin(θ/2), is the straight line between the two ends of the arc.

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