Chord Length Formula

A chord is a straight segment joining two points on a circle. For a central angle θ, its length is c = 2r·sin(θ/2). The longest chord is the diameter, when θ = 180° and c = 2r.

Chord Length formulas

Chord length: $c = 2r\sin\!\left(\dfrac{\theta}{2}\right)$

where r = radius, θ = central angle (degrees), c = chord length

Chord length

c = 2r\sin\!\left(\dfrac{\theta}{2}\right)

Worked example

A 120° chord on a radius-6 circle measures 6√3 ≈ 10.39 units.

c = 2r\sin(\theta/2) = 2\cdot 6\cdot \sin 60^\circ = 12\cdot \tfrac{\sqrt3}{2} = 6\sqrt3 \approx 10.39

Interactive chord length explorer

Drag a slider (or type a value, or use the arrow keys) and watch the diagram, the results, and the full step-by-step substitution update live.

Radius r

Central angle θ θ

Chord length c 10.39units

Step-by-step substitution

chord = 2r · sin(θ/2) = 2 · 6 · sin(60°) = 10.39

Classroom use: chord length formulas

Who it's for: Grades 9–12 · High-school geometry students studying relationships among chords, radii, and central angles.

Learning objectives

After using this page, students will be able to:

Compute chord length from a radius and central angle.
Recognise the diameter as the longest possible chord.
Use the perpendicular from the centre to a chord to reason about its length.

Suggested classroom activity

Sweep the central angle from 0° to 180° and have students describe when the chord grows fastest. Pair this with the isosceles-triangle construction on paper: the explorer supplies the numbers, the construction supplies the reason.

Standards alignment

Aligned to the Common Core State Standards for Mathematics. Codes are given exactly as published so they can be matched against state crosswalks.
Code	Standard
`HSG.C.A.2`	Identify and describe relationships among inscribed angles, radii, and chords.
`8.G.B.7`	Apply the Pythagorean theorem to find unknown side lengths in right triangles.

Free to use and adapt in class under the terms on our license page — no account, no sign-up, no ads. Content last reviewed August 2026.

Related & next steps

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Frequently asked questions

What is the chord length formula in geometry?

For a central angle θ and radius r, the chord length is c = 2r·sin(θ/2). When θ = 180° the chord is a diameter, c = 2r, the longest possible chord.

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