Chord Length Calculator

Calculate the length of a chord.

Result:

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Bridging the Curve: The Chord Length Formula

In the elegant world of geometry, the circle is King. But in the practical world of construction and engineering, we rarely measure "arcs" directly with a tape measure. Instead, we measure straight lines. The straight line that connects any two points on the edge of a circle is called a Chord. Whether you are a carpenter building a curved archway, a machinist checking a hole pattern, or a student solving a trigonometry problem, finding the exact length of this line is essential. Our Chord Length Calculator does the trigonometry for you, converting radius and angle into a precise linear measurement.

This guide explains the geometry behind the tool, derives the formula from scratch, and explores how this simple measurement underpins everything from bridge design to pizza cutting.

What Exactly is a Chord?

A chord is a line segment whose endpoints both lie on the circle.
The Diameter: This is a special chord. It passes through the center and is the longest possible chord in any circle.
The Segment: The area "cut off" by the chord is called a circular segment.

The Pythagorean Derivation

How do we find the length if we only know the Radius (r) and the Central Angle (θ)? We use trigonometry.

  1. Imagine a triangle formed by the center of the circle and the two endpoints of the chord.
  2. This is an Isosceles Triangle because two sides are the Radius (r).
  3. The angle between the two radii is $\theta$.
  4. If we draw a line from the center perpendicular to the chord (the apothem), it splits the triangle into two identical Right-Angled Triangles.
  5. The angle at the top becomes $\theta/2$. The chord is split into two halves ($c/2$).

Now, we apply simpler trigonometry (SOH CAH TOA):
$\sin(\text{angle}) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
$\sin(\theta/2) = \frac{c/2}{r}$

Solving for $c$ gives us the Golden Formula:

Chord Length (c) = 2r × sin(θ/2)

How to Use the Calculator

You only need two values:

  • Radius (r): The distance from the center to the edge. (e.g., 5 meters).
  • Central Angle (degrees): The width of the "slice" of the circle. (e.g., 90 degrees).

Example:
Radius = 10, Angle = 60°.
Math: $2(10) \times \sin(30°)$.
$\sin(30°) = 0.5$.
Result: $20 \times 0.5 = 10$.
(This makes sense because a 60° slices forms an Equilateral Triangle, so calculating side length equals radius!)

Related Concepts

Once you have the chord length, you can calculate other interesting properties:

  • Sagitta (Arc Height): The distance from the center of the chord to the top of the arc. Useful for determining how "deep" an arch is. Formula: $s = r(1 - \cos(\theta/2))$.
  • Apothem: The distance from the center of the circle to the midpoint of the chord.

Real-World Applications

  • Construction: Framing a circular window. You know the width (chord) and height (sagitta), and need to find the Radius to draw the curve.
  • Machining: Bolt hole circles. Calculating the distance between two holes (the chord) to verify patterns.
  • Sports: Calculating the width of goalposts or markings on a curved track.

Conclusion

The chord represents the bridge between the curved world of circles and the linear world of rulers and tape measures. By mastering the relationship between the radius, the angle, and the chord, you can measure and construct complex circular shapes with the precision of a master geometrician.