Chord Length Calculator

Radius (

r

OR Diameter (

D

Central angle (

θ

OR Distance from centre (

d

): OR Chord height / Sagitta (

h

): Result Unit:

Decimal Places: 2 3 5 8

Clear Reset

Introduction

The length of a chord in a circle can be determined from a defining circle dimension - such as the radius $r$ or diameter $D$ - together with a segment-specific parameter including the central angle $θ$ , distance from the centre $d$ , or sagitta $h$ . These measurements describe the geometry of the associated circle segment and allow the chord length and related angular or linear quantities to be derived through standard trigonometric relationships.

What this calculator does

Performs a series of trigonometric and algebraic operations to solve for unknown segment attributes. It requires the input of a circle dimension and exactly one defining segment characteristic. It outputs the chord length, central angle in degrees and radians, apothem, sagitta, arc length, perimeter, and the areas for both the minor and major segments, facilitating a comprehensive analysis of the geometric structure.

Formula used

Calculations rely on the relationship between the radius $r$ and the central angle $θ$ (expressed in radians). The chord length $L$ is derived using the sine of the half-angle. If the distance from the centre $d$ or sagitta $h$ is provided, the Pythagorean theorem and inverse trigonometric functions are applied to establish the missing variables.

Chord Length (L):

L = 2 r sin (\frac{θ}{2})

Arc Length:

s = r θ

Minor Segment Area:

A_{minor} = \frac{1}{2} r^{2} (θ - sin θ)

Major Segment Area:

A_{major} = π r^{2} - A_{minor}

Segment Perimeter:

P = L + s

How to use this calculator

Enter the radius $r$ or diameter $D$ of the circle.
Input exactly one value for the central angle, distance from the centre, or chord height.
Select the preferred unit of measurement and decimal precision.
Execute the calculation to view the results, step-by-step process, and visual representation.

Example calculation

Scenario: Determining the chord length of a circular arc when the radius and sagitta (height of the arc above the chord) are known.

Inputs: Radius $r = 5$ and sagitta $h = 2$ .

Working:

Step 1: Use the chord-sagitta relationship: $L = 2 \sqrt{2 r h - h^{2}}$

Step 2: Substitute the values: $L = 2 \sqrt{2 \times 5 \times 2 - 2^{2}}$

Step 3: Simplify inside the root: $L = 2 \sqrt{20 - 4}$

Step 4: Evaluate the root: $L = 2 \sqrt{16} = 8$

Result: The chord length is 8.00 units.

Interpretation: A sagitta of 2 units in a circle of radius 5 produces a chord 8 units long.

Summary: The radius and sagitta fully determine the geometry of the segment.

Understanding the result

The results describe the spatial boundaries of the segment. The chord length represents the shortest linear distance between two points on the circumference, while the segment area quantifies the space enclosed between that chord and its corresponding arc, revealing the segment's proportion relative to the total circle.

Assumptions and limitations

The calculations assume a perfect Euclidean circle. Input values for radius must be positive, and constraints apply to secondary inputs: the central angle must be strictly between 0 and 360° (0 and $2 π$ radians), and the sagitta cannot exceed the circle diameter.

Common mistakes to avoid

A frequent error is providing multiple secondary inputs simultaneously, which creates an over-determined system. Users should also ensure that the distance from the centre does not equal or exceed the radius, as this would prevent the formation of a chord within the circular boundary.

Sensitivity and robustness

The calculation is stable for most values but exhibits increased sensitivity as the central angle approaches $0$ or $2 π$ . In these extremes, small variations in the radius or height can lead to significant percentage changes in the resulting segment area and chord-to-radius ratio.

Troubleshooting

If an error occurs, verify that the radius is a positive non-zero number. Ensure that the sagitta is greater than zero but less than the diameter. If using degrees, the angle must fall strictly between 0 and 360 to remain mathematically valid for a single segment.

Frequently asked questions

What is a sagitta?

The sagitta is the vertical distance from the centre of the chord to the highest point of the arc, also known as the chord height.

What is the apothem?

The apothem is the perpendicular distance from the centre of the circle to the midpoint of the chord.

How is the arc-to-chord ratio used?

This ratio compares the curved path length to the straight-line distance, indicating the degree of curvature of the segment.

Where this calculation is used

This mathematical modelling is prevalent in advanced geometry and trigonometry courses for studying circular functions. In educational settings, it is used to analyse the properties of sectors and segments, providing a foundation for calculus when exploring limits of polygons inscribed in circles. It is also applied in environmental science and physics to model wave segments or the cross-sectional areas of fluid flow in partially filled pipes, where determining the area of a circular segment is essential for volume flow calculations.

Critical geometric thresholds

When analysing circle segments, certain mathematical boundaries change the behavior of the outputs. Understanding these reference points helps verify your results:

The Semicircle Boundary: When the sagitta $h$ exactly equals the radius $r$ , the central angle is exactly 180° ( $π$ rad). At this precise threshold, the chord transitions into the diameter, the apothem becomes zero, and the minor and major segments are perfectly equal in area.
Minor vs. Major Segments: If the sagitta is less than the radius, you are calculating a minor segment (less than half the circle). If the sagitta is greater than the radius, it forms a major segment, meaning the central angle has exceeded 180° and the apothem extends in the opposite direction.
The Infinitesimal Chord: As the central angle approaches 0°, the arc length and chord length converge toward an identical value. This specific relationship forms the fundamental basis for modern calculus limits.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.