How to Calculate Chord Length

This article is an excerpt from The CMM Handbook

A chord of a circle is a line segment with its endpoints on the circle. The diameter of a circle is a chord that intersects the center of the circle. A line that contains a chord of a circle is known as a secant of that circle. It is possible to calculate a chord's length when the circle's radius and the central angle of the chord are given. We can also calculate a chord's length when we know the circle's radius and distance of the chord to the center of the circle.

Step 1: Define the terms for our calculations. The chord AB of a circle has the points A and B as its endpoints. The center of the circle is the point C and the radius of the circle is r.

Step 2: Use the radius and central angle of the chord to find the chord's length. The central angle of the chord is the angle ACB and will be called c. The length of the chord AB is then given by 2r sin (c/2).

Step 3: Find the chord's length for a circle of radius 3 and central angle of 60 degrees. It is 2r sin (c/2) = 2 x 3 sin (60 degrees/2) = 6 sin (30 degrees) = 6(1/2) = 3.

Step 4: Use the radius and distance of the chord from its center to find the length of the chord. This distance is the length of the line segment d that has the center C and the midpoint of the chord as its endpoints. The length of the chord is then given by 2(r^2 -- d^2)^(1/2).

Step 5: Find the chord's length for a circle of radius 5 that is 3 units from the center. It is 2(r^2 -- d^2)^(1/2) = 2(5^2 -- 3^2)^(1/2) = 2(25 -- 9)^(1/2) = 2(16)^(1/2) = 2 x 4 = 8.