Chord length calculator

What is a chord length calculator?

A chord is a straight line segment whose two endpoints both lie on a circle. The longest chord of a circle is its diameter; every other chord is shorter and is “subtended” by some central angle — the angle formed at the centre by the two radii drawn to the chord’s endpoints.

This calculator finds any one of the three values — chord length, radius, or central angle — when the other two are known. The angle can be entered in degrees or radians, and the radius and chord can be entered in any common unit of length.

Key concepts

• Radius (r) — the distance from the centre of the circle to a point on its boundary.
• Central angle (θ) — the angle formed at the centre of the circle by the two radii drawn to the chord’s endpoints.
• Chord (c) — the straight-line distance between the two endpoints of the arc, cutting across the circle rather than following its curve.
• Diameter — the special case of a chord that passes through the centre. It has length $2r$ and corresponds to a central angle of 180°.

The chord and the arc length describe the same pair of endpoints from two different perspectives: the chord is the shortcut straight across, the arc is the path along the circle.

How does the calculator work?

The chord, the two radii to its endpoints, and the perpendicular dropped from the centre form two congruent right triangles. Half of the chord, the radius, and half of the central angle satisfy

$$\sin\!\left(\tfrac{\theta}{2}\right) = \frac{c/2}{r}$$

which rearranges into the formulas the calculator uses.

Formulas

Chord from radius and central angle:

$$c = 2 r \sin\!\left(\tfrac{\theta}{2}\right)$$

Radius from chord and central angle:

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

Central angle from chord and radius:

$$\theta = 2 \arcsin\!\left(\frac{c}{2r}\right)$$

In degrees, replace $\theta$ with $\theta_{\text{deg}} \cdot \frac{\pi}{180}$, or read the angle directly from the calculator after switching the unit selector.

Worked examples

Example 1: chord from radius and angle

A circle has a radius of 10 cm and a central angle of 60°. The chord cut out by that angle is

$$c = 2 \cdot 10 \cdot \sin(30°) = 2 \cdot 10 \cdot 0.5 = 10 \text{ cm}$$

This is the familiar identity that the chord of a 60° angle equals the radius — the triangle formed is equilateral.

Example 2: chord equals diameter at 180°

For a radius of 5 m and a central angle of 180° (or $\pi$ radians), the chord stretches all the way across the circle:

$$c = 2 \cdot 5 \cdot \sin(90°) = 10 \text{ m}$$

This is the diameter of the circle.

Example 3: radius from chord and angle

A chord 10 cm long is cut by a 60° central angle. The circle’s radius is

$$r = \frac{10}{2 \sin(30°)} = \frac{10}{1} = 10 \text{ cm}$$

Example 4: angle from chord and radius

A chord 10 cm long is drawn in a circle of radius 10 cm. The central angle is

$$\theta = 2 \arcsin\!\left(\frac{10}{20}\right) = 2 \cdot 30° = 60°$$

Example 5: chord of a quarter circle

For a 90° angle on a circle of radius 1, the chord is $c = 2 \sin(45°) = \sqrt{2} \approx 1.4142$, while the arc length of the same angle is $\pi/2 \approx 1.5708$. The arc is always slightly longer than the chord.

Practical uses

• Engineering — laying out belts and pulleys, where the straight-line distance between contact points on two wheels is a chord of each wheel.
• Architecture and carpentry — measuring across an arch or a curved window, where the chord gives the span and the arc length gives the material needed along the curve.
• Surveying — fixing positions on the ground from circular reference points; chord measurements are easier to mark than arcs.
• Astronomy — computing the apparent diameter of distant bodies, where the chord across a circular cross-section corresponds to the observed extent.
• Geometry and trigonometry — the chord/angle relationship is one of the original definitions of the sine function and still appears in circle sector and segment calculations.

Notes

• The chord can never be longer than the diameter ($c \le 2r$). If you enter a chord longer than that, the angle is undefined and the calculator returns no result.
• A 0° angle gives a chord of 0 — the endpoints coincide.
• A 180° angle gives the diameter; angles larger than 180° wrap around and give the same chord as their supplement (e.g. 200° and 160° yield identical chords).
• When solving for the radius from a chord and an angle, the angle cannot be 0; when solving for the angle, the radius cannot be 0.
• Radius and chord share units: switching the unit selector reconverts the result automatically.