Chord Calculator - Find Chord Length in Circle

The primary chord length formula is Chord = 2r sin(θ/2), where r is the circle's radius and θ (theta) is the central angle in radians. This elegant relationship comes directly from splitting the isosceles triangle formed by two radii and the chord into two right triangles.

Each right triangle has hypotenuse r, an angle of θ/2 at the center, and opposite side equal to half the chord. By the sine ratio: half-chord = r sin(θ/2), so the full chord = 2r sin(θ/2). This derivation is why the formula appears so often in geometry textbooks and engineering references.

All three chord length formulas

• From the central angle: Chord = 2r × sin(θ/2) — use when you know the angle subtended at the center.
• From the sagitta (arc height): Chord = 2√(h(2r − h)) — use when you know the perpendicular distance from the chord's midpoint to the arc.
• From the distance from center to chord: Chord = 2√(r² − d²) — use when you know how far the chord sits from the circle's center.

All three formulas are mathematically equivalent. Given any two of: radius, central angle, sagitta, or center-distance, you can derive the rest. Our chord calculator handles all three inputs and outputs every related value automatically.

Follow these steps to calculate chord length using the 2r sin(θ/2) formula:

1. Identify the radius — measure or read the circle's radius r. If you only have the diameter, divide by 2.
2. Find the central angle — this is the angle θ formed at the center by two radii drawn to the chord's endpoints. Convert degrees to radians if needed: radians = degrees × π/180.
3. Apply the formula — Chord = 2r × sin(θ/2). Divide the central angle by 2, take its sine, then multiply by 2r.
4. Verify — the chord must be ≤ the diameter (2r). A 180° central angle gives a chord equal to the diameter; a 60° angle gives a chord equal to the radius.

Worked example: radius 10, central angle 90°

Convert 90° to radians: θ = 90 × π/180 = π/2 ≈ 1.5708 rad. Half-angle = π/4. sin(π/4) = √2/2 ≈ 0.7071. Chord = 2 × 10 × 0.7071 = 14.142 units. The distance from center to this chord is d = r cos(θ/2) = 10 × cos(45°) ≈ 7.071 units.

The perpendicular distance from the center of a circle to a chord is given by d = r cos(θ/2). This distance is also called the apothem of the chord. It always meets the chord at its midpoint, forming a right angle — a key theorem in circle geometry.

This relationship connects directly to the chord length formula. Since chord = 2r sin(θ/2) and d = r cos(θ/2), we also get: chord² + (2d)² = (2r)², which is the Pythagorean theorem applied to the chord, the perpendicular bisector, and the diameter.

Why does the perpendicular from center bisect the chord?

Draw two radii to the chord endpoints. These form an isosceles triangle (both sides = r). The perpendicular from the apex (center) to the base (chord) of an isosceles triangle always bisects the base. This is why d = r cos(θ/2) and half-chord = r sin(θ/2) are guaranteed by symmetry.

If you know the distance d but not the angle, you can find the chord length with Chord = 2√(r² − d²). Our calculator handles this input type directly — just select "Center Distance" and enter your values.

The sagitta (also called the arc height or versine) is the perpendicular distance from the midpoint of a chord to the arc. It equals h = r − d = r(1 − cos(θ/2)). Given the sagitta h and radius r, the chord length is:

Chord = 2√(h × (2r − h))

This formula is especially useful in construction and manufacturing, where the sagitta is easy to measure physically — place a straight edge across an arc and measure the gap at the center. Use our arc length calculator to convert between arc length, chord length, and sagitta for complete circular segment analysis.

How to find the radius from chord length and sagitta

Rearranging the sagitta-chord formula gives r = (c²/(8h)) + h/2, where c is the chord length and h is the sagitta. This is invaluable when reconstructing a circle from a curved fragment — common in archaeology, sheet-metal work, and surveying. Use our radius calculator for direct radius computations.

A circle chord calculator has broad applications across engineering, construction, and science. Here are the most common uses:

Understanding these chord theorems helps you solve geometry problems and verify calculator results:

• Perpendicular bisector theorem: A perpendicular from the center of a circle to a chord bisects the chord. Conversely, the perpendicular bisector of any chord passes through the center.
• Equal chords, equal distances: Chords that are equal in length are equidistant from the center. Longer chords are always closer to the center.
• Inscribed angle theorem: An inscribed angle is half the central angle that subtends the same arc. So the inscribed angle = θ/2.
• Intersecting chords theorem: If two chords intersect inside a circle, the products of their segments are equal: AE × EB = CE × ED.
• Diameter is the longest chord: When θ = 180°, the chord equals the diameter 2r, which is the maximum possible chord length.
• Chord-arc relationship: For small central angles, chord length ≈ arc length. The difference grows as the angle increases.

For deeper exploration of circle geometry, visit Khan Academy's circle geometry course, which covers chord theorems with interactive practice problems.

Our chord length calculator offers four input modes to match whatever information you have:

• Central Angle mode: Enter the radius and the central angle (degrees or radians) to find chord length via 2r sin(θ/2).
• Sagitta mode: Enter the radius and the arc height to find chord length via 2√(h(2r−h)).
• Center Distance mode: Enter the radius and the perpendicular distance from center to chord to find chord length via 2√(r²−d²).
• Chord Length mode: Enter a known chord length and radius to reverse-calculate the central angle, sagitta, and distance from center.

Every calculation returns the full set of related values: chord length, central angle (degrees and radians), inscribed angle, sagitta, distance from center, arc length, segment area, and sector area — with a complete step-by-step derivation so you can follow the math or include it in your homework or report.