Chord Length Calculator

How the Chord Length Calculator Works

The Chord Length Calculator determines the length of a chord in a circle using either the central angle, chord height, or apothem. This tool is essential for students of geometry and professionals in engineering, architecture, manufacturing, and land surveying who work with arcs and circular segments.

With the calculator, you also receive related measurements like chord height, apothem, arc length, and sector area, offering a complete geometric analysis of any circular segment.

Formula

When using the central angle (α) in degrees, the chord length c is calculated as:

• Chord Length (c) = 2 × r × sin(α / 2)

Related values:

• Chord Height (h) = r - r × cos(α / 2)

• Apothem = r × cos(α / 2)

• Arc Length = r × α × (π / 180)

• Sector Area = (π × r² × α) / 360

These formulas are rooted in trigonometry and represent key geometric relationships between the circle's radius and its internal angles.

Example Calculation

Given:

• Radius (r) = 2 cm

• Central angle (α) = 3°

Calculations:

• Chord Length: 2 × 2 × sin(3° ÷ 2) ≈ 0.1047 cm

• Chord Height: 2 - 2 × cos(1.5°) ≈ 0.0007 cm

• Apothem: 2 × cos(1.5°) ≈ 1.9993 cm

• Arc Length: 2 × 3 × (π / 180) ≈ 0.1047 cm

• Sector Area: (π × 2² × 3) / 360 ≈ 0.1047 cm²

This shows how a small central angle produces a short chord and near-flat arc.

Geometric Relationships

Additionally:

• Radius = h + apothem

• Triangle Area = ½ × chord × apothem

• Segment Area = Sector Area - Triangle Area

These relationships help compute areas, dimensions, and boundaries of circular segments and arcs.

Applications

The Chord Length Calculator is widely used in:

• Architecture: Arched doorways, windows, and curved ceilings

• Engineering: Bridge arches, tunnel designs, and dome structures

• Manufacturing: CNC cutting, curved components, mold shaping

• Surveying: Mapping terrain and measuring curved property lines

Knowing the chord's exact length and height aids in precision fabrication and construction.

Difference Between Chord and Arc

• Chord: A straight line between two points on a circle

• Arc: The curved path along the circle between those two points

While an arc measures distance along the curve, a chord measures the direct straight-line distance.

How does the chord height affect calculations?

The chord height (h) gives the maximum distance from the chord to the arc. It increases with larger central angles and is critical when calculating segment areas and structural clearances.

If you have the chord height instead of the angle, the calculator can still find the chord using reverse trigonometry.

Can the calculator use radians?

Yes, when angle input is in radians, formulas adjust accordingly:

• Chord: c = 2r × sin(θ / 2)

• Arc Length: L = r × θ

• Sector Area: A = ½ × r² × θ

Be sure to choose the correct unit before calculation.

Can any length unit be used?

Absolutely. As long as the input values (radius and angles) use consistent units, the results (lengths and areas) will match — centimeters, meters, inches, etc.

What if the angle is very small?

Very small angles produce:

• Very short chords

• Very shallow chord heights

• Almost flat arcs

These cases often arise in precision manufacturing or engineering analysis.