How to Calculate Chord Length

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A chord is a line segment connecting any two points on the circumference of a circle. The circle's diameter, the line segment through the center, is also its longest chord. You can calculate the length of a chord from the length of the radius and the angle made by lines connecting the circle's center to the two ends of the chord. You can also calculate chord length if you know both the radius and the length of the right bisector, which is the distance from the center of the circle to the center of the chord.

TL;DR (Too Long; Didn't Read)

You can calculate chord length of a circle if you know the radius and one of two other variables. One variable is the length of a perpendicular line from the chord to the center of the circle. The other is the angle formed by two radius lines that touch the intersection points of the chord and the circumference of the circle.

Basic Strategy for Calculating Chord Length

The trigonometric procedure for calculating chord length starts by extending radius lines to each point at which the chord intersects the circumference of the circle. This creates a triangle with one apex at the center of the circle and an apex at each of the intersection points. If you extend a perpendicular line from the chord to the center of the circle, it will bisect the angle of that apex and create two right triangles on either side of the chord. If the entire angle is ​θ​ (theta), the angle on either side of the bisection line is ​θ​/2.

You can now set up an equation that relates the chord length (​c​) to the radius (​r​) and the angle between the two radius lines (​θ​). Because half the chord line (​c​/2) forms the opposing line in a right-angle triangle, and ​r​ forms the hypotenuse, the following is true:

\sin \bigg(\frac{θ}{2}\bigg) = \frac{c/2}{r}

Solving for ​c​:

c = \text{ chord length }= \sin \bigg(\frac{θ}{2}\bigg)

If you know the radius of the circle and can measure the angle ​θ​, you have all you need to calculate chord length.

Calculating Chord Length When You Can't Measure Angle

In practice, it can be difficult to measure the angle formed by the radius lines. For example, you may be planning to erect a fence that extends from one point on a circular plot of land to another, and you need to know how long the fence has to be. You can still use trigonometry to find the answer if you know the radius and can measure the distance from the chord to the center of the circle. As long as the line is perpendicular to the chord, it divides it in two and forms a right triangle. If the length of that line is ​l​, the Pythagorean Theorem tells you that

l^2 + \bigg(\frac{c}{2}\bigg)^2 = r^2

Solving for ​c​:

c = 2 × \sqrt{r^2 - l^2}

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