Chord Length Calculator

What Is a Circle Chord?

In Euclidean geometry, a chord of a circle is a straight line segment whose endpoints both lie on the circle's perimeter. If a chord passes directly through the center of the circle, it is called the diameter, which is the longest possible chord in any given circle.

Chords are fundamental elements in trigonometry, structural architecture, civil engineering, optics, and navigation. When a chord connects two points on a circular curve, it divides the circle into two circular segments and forms an isosceles triangle with the center of the circle. You can easily analyze related circle properties using our Circle Calculator, Circle Arc Calculator, and Circle Segment Calculator.

Chord Length Formulas

Depending on which dimensions of the circle and chord are known, you can compute the chord length $c$, radius $r$, central angle $\theta$, sagitta (chord height $h$), or apothem (center distance $d$) using the following standard geometric equations:

1. From Radius ($r$) and Central Angle ($\theta$)

When you know the radius $r$ and the subtended central angle $\theta$ (in radians or degrees):

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

Where:

• $c$ is the chord length
• $r$ is the circle radius
• $\theta$ is the central angle subtending the chord

2. From Radius ($r$) and Sagitta / Chord Height ($h$)

The sagitta $h$ is the vertical distance from the center of the chord to the arc of the circle. Using the Pythagorean theorem:

$$c = 2\sqrt{2rh - h^2} = 2\sqrt{h(2r - h)}$$

3. From Radius ($r$) and Apothem ($d$)

The apothem $d$ is the shortest perpendicular distance from the center of the circle to the chord ($d = r - h$):

$$c = 2\sqrt{r^2 - d^2}$$

4. Finding Circle Radius ($r$) from Chord Length ($c$) and Height ($h$)

In carpentry, stone masonry, and curved beam design, builders often measure the span (chord length $c$) and rise (sagitta $h$) to find the unknown radius:

$$r = \frac{c^2 + 4h^2}{8h} = \frac{h}{2} + \frac{c^2}{8h}$$

Related Geometric Properties of a Chord

Step-by-Step Calculation Example

Suppose a circular arch has a radius of $r = 10\text{ meters}$ and the central angle subtended by the arch is $\theta = 60^\circ$.

1. Convert angle to radians: $\theta_{\text{rad}} = 60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \approx 1.0472\text{ rad}$.
2. Compute chord length ($c$): $$c = 2 \times 10 \times \sin\left(\frac{60^\circ}{2}\right) = 20 \times \sin(30^\circ) = 20 \times 0.5 = 10\text{ meters}$$
3. Compute apothem ($d$): $$d = 10 \times \cos(30^\circ) = 10 \times \frac{\sqrt{3}}{2} \approx 8.6603\text{ meters}$$
4. Compute sagitta / rise ($h$): $$h = r - d = 10 - 8.6603 = 1.3397\text{ meters}$$
5. Compute segment area ($A_{\text{seg}}$): $$A_{\text{seg}} = \frac{1}{2} \times 10^2 \times (1.0472 - \sin(60^\circ)) = 50 \times (1.0472 - 0.8660) \approx 9.0586\text{ m}^2$$