Calculating Chord Length: A Complete Geometry Guide
In the study of circles, few measurements are as fundamental yet as frequently misunderstood as the chord length. A chord is simply a straight line segment whose endpoints both lie on the circle's circumference. But unlike the radius, which connects the center to the edge, or the diameter, which passes through the center, a chord can exist at any orientation and position within a circle. Understanding how to calculate its length is essential for students, engineers, architects, and anyone working with circular designs, from pie charts to planetary orbits. This article breaks down the most reliable methods for determining chord length, rooted in geometry and ready for practical application.
The Geometry of a Chord
Before applying formulas, it helps to visualize the components involved. Every chord has a corresponding central angle—the angle formed at the circle's center by lines drawn to the chord's endpoints. Now, additionally, the perpendicular distance from the circle's center to the chord (often called the sagitta or apothem when dealing with segments) is key here in many calculations. Still, one of the most useful properties in circle geometry is that the perpendicular from the center to a chord always bisects the chord into two equal segments and also bisects the central angle. This symmetry is the foundation for the three primary calculation methods discussed here.
Calculating Chord Length When the Central Angle Is Known
The most direct approach involves the radius of the circle and the measure of the central angle that subtends the chord. If a circle has radius $r$ and the central angle $\theta$ (in radians) is known, the chord length $L$ is given by:
$L = 2r \sin\left(\frac{\theta}{2}\right)$
When the angle is measured in degrees, the formula adjusts to:
$L = 2r \sin\left(\frac{\theta \cdot \pi}{360}\right)$
This formula derives from the law of cosines applied to the isosceles triangle formed by the two radii and the chord. The triangle has two sides of length $r$ and an included angle $\theta$. The chord serves as the base, and splitting that triangle in half creates two right triangles, each with hypotenuse $r$, angle $\theta/2$, and opposite side equal to half the chord length And that's really what it comes down to..
