Chord Length Calculator: Understanding Circle Chords and Formulas

A chord is a fundamental part of a circle, connecting any two points on its circumference. Whether you're studying geometry, working on a design project, or just curious about circle properties, knowing how to calculate its length is a useful skill. This guide and our Chord Length Calculator will help you master the formulas and understand the concepts behind finding a chord's length.

Explanation

A chord is a straight line segment whose endpoints both lie on the circumference of a circle. The longest chord in any circle is its diameter, which passes through the center. There are two primary ways to calculate the length of a chord, depending on the information you have: **1. Using the Radius and Central Angle:** If you know the radius (r) of the circle and the central angle (θ) that the chord subtends (the angle formed at the center of the circle by the two radii connected to the chord's endpoints), you can use trigonometry. The formula is derived from splitting the isosceles triangle formed by the two radii and the chord into two right-angled triangles. **2. Using the Radius and Sagitta:** The sagitta (s) is the height of the circular segment, which is the perpendicular distance from the midpoint of the chord to the arc. This measurement is often used in engineering and architecture. The formula for chord length using the sagitta is derived from the Pythagorean theorem.

Formula

The two main formulas for calculating chord length are: **Using Radius (r) and Central Angle (θ):**
Chord Length = 2 * r * sin(θ/2) *Note: The central angle (θ) must be in radians for this formula. If you have it in degrees, convert it by multiplying by π/180.* **Using Radius (r) and Sagitta (s):**
Chord Length = 2 * sqrt(s * (2r - s))

Example

Let's walk through an example to see how these formulas work. Suppose you have a circle with a radius of 15 inches. A particular chord in this circle creates a central angle of 90 degrees. First, convert the angle to radians: 90 degrees * (π/180) = π/2 radians. Now, apply the formula: Chord Length = 2 * 15 * sin((π/2)/2) Chord Length = 30 * sin(π/4) Chord Length = 30 * (√2 / 2) Chord Length = 15√2 ≈ 21.213 inches If, instead, you knew the radius was 15 inches and the sagitta was 3 inches, you would use the second formula: Chord Length = 2 * sqrt(3 * (2 * 15 - 3)) Chord Length = 2 * sqrt(3 * (30 - 3)) Chord Length = 2 * sqrt(3 * 27) Chord Length = 2 * sqrt(81) Chord Length = 2 * 9 = 18 inches

How to use the related calculator

Our Chord Length Calculator makes these calculations straightforward. Simply enter the radius of your circle in the 'Radius (r)' field. Then, choose one of the two methods: 1. **Using Central Angle:** Enter the angle in either 'Central Angle (θ) in Degrees' or 'Central Angle (θ) in Radians'. You only need to fill in one of these angle fields. 2. **Using Sagitta:** Alternatively, enter the sagitta (height of the circular segment) in the 'Sagitta (s)' field. The calculator will instantly display the calculated chord length. Make sure to only provide inputs for one calculation method (angle OR sagitta) along with the radius to get an accurate result.

Try the related calculator

FAQ

Related articles

About the author

Muhammad Ali. Muhammad Ali is a full-stack developer and founder of KruskalCode. He builds SaaS platforms and automation systems with React and Laravel, and helps teams ship fast, scalable tools.

KruskalCode•GitHub•Email

Need a custom calculator, dashboard, or automation workflow? Reach out to KruskalCode.