Chord Length Calculator Online | Circle Geometry Tool for Students & Professionals

What is a Chord and Chord Length?

Definition and basic properties of chords
Relationship between chords and circle geometry
Mathematical significance in various applications

A chord is a straight line segment that connects any two points on the circumference of a circle. It represents one of the fundamental elements in circle geometry, alongside radius, diameter, and tangent lines.

Key Properties of Chords

The length of a chord depends on two primary factors: the radius of the circle and the central angle it subtends. The central angle is the angle formed at the center of the circle by drawing lines from the center to both endpoints of the chord.

The diameter of a circle is actually the longest possible chord, as it passes through the center and subtends a central angle of 180° (π radians). Any other chord in the same circle will be shorter than the diameter.

Mathematical Relationship

The fundamental formula for chord length is c = 2r × sin(θ/2), where c is the chord length, r is the radius, and θ is the central angle in radians. This formula derives from the application of trigonometry to circle geometry.

Basic Chord Examples

In a circle with radius 10 units, a central angle of 60° creates a chord length of 10 units
The diameter is always the longest chord, with a central angle of 180°
Equal chords in the same circle subtend equal central angles

Step-by-Step Guide to Using the Chord Length Calculator

Selecting the appropriate calculation type
Input validation and unit considerations
Interpreting and verifying results

Our chord length calculator offers three calculation modes: finding chord length, finding radius, and finding central angle. Each mode requires two known values to calculate the third unknown value.

Calculation Types

When finding chord length, input the radius and central angle. The calculator uses the formula c = 2r × sin(θ/2) to determine the chord length. This is the most common calculation type for engineering and design applications.

For finding radius, input the chord length and central angle. The calculator rearranges the formula to r = c / (2 × sin(θ/2)). This calculation is useful when you have physical measurements of a chord and need to determine the circle's size.

Angle Unit Selection

The calculator supports both degrees and radians. Degrees are more intuitive for most users, while radians are often required for advanced mathematical calculations. The conversion is automatic based on your selection.

Remember that angles must be positive and less than 360° (2π radians) for valid chord calculations. The chord length cannot exceed the diameter of the circle.

Usage Guidelines

Always verify that your chord length is less than 2 × radius
Use degrees for practical applications, radians for theoretical work
Cross-check results using different calculation modes when possible

Real-World Applications of Chord Length Calculations

Architecture and structural engineering
Manufacturing and design applications
Navigation and surveying

Chord length calculations are essential in architecture for designing arches, domes, and curved structural elements. Architects use these calculations to determine the dimensions of curved beams and the spacing of support structures.

Engineering Applications

In mechanical engineering, chord lengths are crucial for gear design, cam profiles, and curved machinery components. The calculations help engineers ensure proper fit and function of rotating parts and curved surfaces.

Bridge construction frequently involves chord calculations for arch bridges and suspension bridge cables. The precise calculation of chord lengths ensures structural integrity and load distribution.

Manufacturing and Design

Manufacturing processes often require chord length calculations for cutting curved materials, designing circular saw blades, and creating curved templates. CNC programming relies on these calculations for precise curved cuts.

In the automotive industry, chord calculations are used for designing curved body panels, wheel specifications, and aerodynamic components.

Industry Applications

Gothic cathedral arches use precise chord calculations for structural stability
Automotive wheel rim design requires chord length calculations for spoke placement
Satellite dish parabolas use chord geometry for optimal signal reception

Common Misconceptions and Correct Methods

Avoiding calculation errors and misunderstandings
Proper unit handling and conversions
Verification techniques and quality checks

One common misconception is that chord length increases linearly with central angle. In reality, the relationship is trigonometric, following the sine function. Small angle changes can have significant effects on chord length.

Unit Consistency

Always ensure units are consistent throughout calculations. Mixing degrees and radians, or different length units, can lead to significant errors. Our calculator handles unit conversions automatically, but manual calculations require careful attention.

Another common error is confusing arc length with chord length. The arc length follows the circumference of the circle, while the chord is the straight-line distance. Chord length is always shorter than the corresponding arc length.

Validation Methods

Always verify that calculated chord lengths are physically possible. The chord cannot be longer than the diameter, and very small central angles should produce correspondingly small chord lengths.

Use the geometric relationship that equal chords subtend equal central angles to cross-check calculations. If two chords have the same length in the same circle, their central angles must be equal.

Verification Examples

A 180° central angle always produces a chord equal to the diameter
Very small angles (< 10°) have chord lengths approximately equal to arc lengths
In any circle, the longest possible chord is the diameter

Mathematical Derivation and Advanced Examples

Trigonometric foundation of chord formulas
Complex calculation scenarios
Integration with other geometric concepts

The chord length formula c = 2r × sin(θ/2) derives from the application of the law of cosines to an isosceles triangle. When we draw radii to both endpoints of a chord, we create an isosceles triangle with two sides of length r and an angle θ between them.

Trigonometric Derivation

Using the law of cosines: c² = r² + r² - 2r·r·cos(θ) = 2r²(1 - cos(θ)). Using the identity 1 - cos(θ) = 2sin²(θ/2), we get c² = 4r²sin²(θ/2), which simplifies to c = 2r·sin(θ/2).

This derivation shows why the formula involves the sine of half the central angle, not the full angle. The half-angle appears because we're effectively working with two right triangles formed by dropping a perpendicular from the center to the chord.

Advanced Applications

In coordinate geometry, chord length can also be calculated using the distance formula when the endpoints are known: c = √[(x₂-x₁)² + (y₂-y₁)²]. This approach is useful in computer graphics and CAD applications.

For multiple chords in the same circle, the relationship between chord lengths and their distances from the center follows the formula: d = √(r² - (c/2)²), where d is the distance from center to chord.

Mathematical Examples

In a unit circle (r=1), a 90° angle produces a chord length of √2 ≈ 1.414
The golden ratio appears in regular pentagon chord calculations
Hexagon inscribed in a circle has all chords equal to the radius

Basic Chord Length

Find Radius from Chord

Central Angle Calculation

Radians Example