The Circle Chord Calculator finds a straight chord length, sagitta, central angle, minor arc length, and related radius or diameter values from two measurements that you already know.
This guide specifically explains this calculator because the calculator page is intentionally short and easy to scan. It identifies the exact What do you know? choices, field labels, units, output wording, formulas, validation rules, diagram, working, copy, clear, and history controls. Use the live calculator for the number, then use this guide to interpret the geometry and verify the result.
What is a chord?
A chord is the straight line segment joining two points on a circle. If the endpoints are called A and B, the chord is written as AB or c. It is not the curved boundary between A and B. The calculator's main output is labelled Chord length and uses the selected linear unit.
For a partial circle, the corresponding minor arc follows the circumference from A to B. It is longer than the straight chord for an ordinary angle between 0° and 180°. The calculator reports this separate output as Minor arc length, so do not substitute one for the other when a problem asks for a straight span.
What is the central angle?
The central angle is the angle with its vertex at the centre of the circle. In the calculator, the exact field label is Central angle (°), which means the input is in degrees. The output is labelled Central angle and is followed by the same angle in parentheses as a radian value.
The two radii from the centre O to the endpoints A and B form angle ∠AOB. The live calculator requires a positive angle below 360° in the modes that use a central angle. A 90° angle describes a quarter of a full turn; it is not the angle of an arbitrary triangle vertex unless that vertex is at the circle centre.
What is sagitta or chord height?
Sagitta, also called chord height, is the perpendicular distance from the midpoint of a chord to the corresponding minor arc. The page uses the exact field label Sagitta / chord height and the exact result label Sagitta / height. In the diagram, the midpoint is marked M and the perpendicular segment is OM.
For a radius r and chord length c, the calculator uses the minor-segment height measured toward the arc:
h = r − √(r2 − (c/2)2)
Because this tool treats the input as the minor chord geometry, a supplied sagitta must be positive and smaller than the radius. A sagitta equal to or greater than the radius produces the calculator's error message Sagitta must be smaller than the radius.
What are radius, diameter, and minor arc length?
The radius, written as r, is the distance from the centre O to the circle. The diameter is the full width through the centre, so d = 2r. The result panel reports both Radius and Diameter in the chosen unit, even when only one of them was entered.
Minor arc length is the curved distance along the smaller arc between the chord endpoints. If the central angle is θ degrees, the calculator uses the fraction θ/360 of a complete circumference. This is a linear result, just like chord length, radius, diameter, and sagitta.
Why does the calculator use π?
π, pronounced “pi,” is the constant ratio of a circle's circumference to its diameter. Its decimal expansion begins 3.14159265359 and continues without terminating. The calculator uses the built-in mathematical value of π instead of asking you to enter it.
Pi appears in the minor-arc formula because a full circle has circumference 2πr. The central-angle fraction selects the required part:
s = 2πr(θ/360)
The calculator converts the degree angle to radians for trigonometry. The conversion is θrad = θπ/180. The radian result displayed in parentheses is useful when checking the sine or arcsine relationship in another calculation.
Which “What do you know?” option should you select?
The calculator's first control is labelled exactly What do you know?. It has five choices. Each choice changes the two numeric field labels and their placeholders, so always read the fields again after changing the menu.
| Exact option | Field 1 | Field 2 | What the calculator derives | Key condition |
|---|---|---|---|---|
| Radius and central angle | Radius Placeholder: Enter radius | Central angle (°) Placeholder: Enter central angle (°) | Chord, diameter, sagitta, central angle, radians, and minor arc | Angle must be greater than 0° and less than 360° |
| Radius and sagitta (chord height) | Radius Placeholder: Enter radius | Sagitta / chord height Placeholder: Enter sagitta / chord height | Chord, diameter, central angle, radians, and minor arc | Sagitta must be smaller than radius |
| Radius and chord length | Radius Placeholder: Enter radius | Chord length Placeholder: Enter chord length | Central angle, radians, diameter, sagitta, and minor arc | Chord cannot be longer than the diameter |
| Chord length and central angle | Chord length Placeholder: Enter chord length | Central angle (°) Placeholder: Enter central angle (°) | Radius, diameter, sagitta, and minor arc | Angle must be less than 360° |
| Diameter and chord length | Diameter Placeholder: Enter diameter | Chord length Placeholder: Enter chord length | Radius, central angle, radians, sagitta, and minor arc | Chord cannot be longer than the diameter |
What are the unit choices?
The second menu is labelled exactly Unit. Its choices are mm, cm, m, in, and ft. The page initially selects cm. Enter both length measurements in the same unit; the calculator does not convert a mixed pair of units for you.
| Selected Unit | Chord, radius, diameter, sagitta, and minor arc | Angle outputs |
|---|---|---|
| mm | mm | degrees and radians |
| cm | cm | degrees and radians |
| m | m | degrees and radians |
| in | in | degrees and radians |
| ft | ft | degrees and radians |
What formulas does the Circle Chord Calculator use?
Chord from radius and central angle
When Radius and central angle is selected, the calculator uses:
c = 2r sin(θ/2)
The angle is entered in degrees, but the trigonometric calculation uses its radian equivalent. The result is the straight distance AB, not the curved arc.
Chord from radius and sagitta
When Radius and sagitta (chord height) is selected, the calculator uses the right-triangle relationship:
c = 2√(2rh − h2)
It then finds the central angle from θ = 2 arcsin(c/(2r)). The input restriction h < r keeps the square-root expression in the intended minor-chord geometry.
Central angle from radius and chord
When Radius and chord length is selected, the calculator reverses the chord formula:
θ = 2 arcsin(c/(2r))
The result is converted from radians to degrees for the displayed Central angle. The chord must not exceed 2r, the diameter.
Radius from chord and central angle
When Chord length and central angle is selected, the calculator solves for the radius:
r = c/[2 sin(θ/2)]
The page specifically rejects an angle of 360° or more with Use a central angle less than 360° for a chord. This prevents a zero sine denominator and keeps two distinct chord endpoints in view.
Radius from diameter and chord
When Diameter and chord length is selected, the first step is simply:
r = d ÷ 2
The calculator then checks the chord against the recovered diameter, finds the central angle with the arcsine formula, and derives the remaining values.
Minor arc length and sagitta
After the mode-specific calculation, every valid result uses:
s = 2πr(θ/360)
h = r − √(r2 − (c/2)2)
The calculator formats displayed numbers to at most four decimal places and removes unnecessary trailing zeroes. The underlying calculation retains more precision than the visible result.
How to use the Circle Chord Calculator
Open the Circle Chord Calculator and use the controls in this order:
- Select the correct pair under What do you know?.
- Read the two generated labels, such as Radius, Chord length, Sagitta / chord height, or Central angle (°).
- Enter only positive numeric values in the two fields. Do not type the unit or a degree symbol into the number field.
- Select mm, cm, m, in, or ft under Unit.
- Click the exact button labelled Click to calculate chord.
- Read Chord length, Radius, Diameter, Central angle, Sagitta / height, and Minor arc length in the result panel.
If either value is missing, zero, negative, or not numeric, the page reports Enter two values greater than zero. A chord longer than its diameter produces The chord cannot be longer than the diameter. Changing the mode redraws the fields and clears the current calculation.
Worked example using every input option
Use the same quarter-circle geometry in all five rows: radius 5 cm, central angle 90°, chord length 7.071067812 cm, sagitta 1.464466094 cm, and minor arc length 7.853981634 cm. Select cm under Unit each time and click Click to calculate chord.
| Exact selection under “What do you know?” | First field | Second field | Expected geometry |
|---|---|---|---|
| Radius and central angle | Radius = 5 | Central angle (°) = 90 | Chord about 7.0711 cm; sagitta about 1.4645 cm |
| Radius and sagitta (chord height) | Radius = 5 | Sagitta / chord height = 1.464466094 | Chord about 7.0711 cm; angle about 90° |
| Radius and chord length | Radius = 5 | Chord length = 7.071067812 | Central angle about 90°; sagitta about 1.4645 cm |
| Chord length and central angle | Chord length = 7.071067812 | Central angle (°) = 90 | Radius about 5 cm; diameter about 10 cm |
| Diameter and chord length | Diameter = 10 | Chord length = 7.071067812 | Radius about 5 cm; central angle about 90° |
For the first row, the main substitution is:
c = 2 × 5 × sin(90°/2) = 7.0711 cm
The remaining displayed values should be approximately:
| Actual output label | Formula or relationship | Quarter-circle result |
|---|---|---|
| Chord length | c = 2r sin(θ/2) | 7.0711 cm |
| Radius | Known or recovered value | 5 cm |
| Diameter | d = 2r | 10 cm |
| Central angle | Input or recovered angle | 90° |
| Radian value | 90π/180 | 1.5708 radians |
| Sagitta / height | h = r − √(r2 − (c/2)2) | 1.4645 cm |
| Minor arc length | s = 2πr(90/360) | 7.854 cm |
Small differences in the last displayed digit are expected when a long chord or sagitta is rounded before it is entered. The first mode is the cleanest reference because the radius and angle are entered directly; the other four modes should reproduce that geometry within the calculator's four-decimal display.
How the diagram, steps, copy, clear, and history controls work
- Click to visualize it opens the calculator's diagram after a valid result. It shows the circle, chord AB, centre O, endpoints A and B, midpoint M, the radii, the perpendicular relationship OM ⊥ AB, the chord length, sagitta, and central angle. The page warns that the diagram is a visual aid; use the numeric result for exact measurements.
- Click to show steps reveals the mode-specific working followed by the sagitta and minor-arc formulas. While the working is open, the button changes to Hide steps.
- Click to copy result copies a compact line containing Chord length, Radius, and Sagitta. After a successful copy, its label temporarily changes to Copied. Review the full result panel first because the copied line does not include every displayed output.
- Clear empties both current fields, removes the error and result, hides the steps and diagram, and disables the visualization and copy controls. Use it before starting a separate problem.
- Click to view history opens the Recent calculations panel. The panel includes Clear history and says that recent calculations appear on this device. A saved entry records the mode, two input values, unit, result label, and time. The calculator retains up to ten entries and removes an exact duplicate of a prior mode/value/unit combination.
Where Do We Use Chord Length, Sagitta, and Minor Arc Length in Real Life?
The outputs describe different parts of the same circular geometry. A straight span, a rise above that span, and a curved edge length should not be treated as interchangeable measurements.
| Calculator output | Distinct real-life use |
|---|---|
| Chord length | Set the straight opening between two points on a circular window, arch, tank wall, wheel rim, bridge profile, or fabricated curved panel. |
| Sagitta / height | Specify how far an arch, curved plate, roof member, roadway profile, or inspection template rises above its straight chord. |
| Minor arc length | Estimate material needed along a curved edge, the length of a rail following a circular path, or the boundary distance around part of a wheel or pipe. |
| Radius | Locate the centre of a curve, set a compass or cutting tool, or check clearance around a rotating circular component. |
| Diameter | Check the full width of a wheel, lid, pipe opening, circular cutout, or round panel against available space. |
| Central angle | Describe the amount of a circular turn occupied by a chord, such as a camera sweep, curved layout, sector boundary, or rotating mechanism. |
| Radians | Pass the angular result to trigonometry, physics, robotics, or software that requires radian measure even though the calculator accepts degrees. |
When this calculator is not the right tool
This calculator is for one chord and its related minor-arc geometry. If the problem asks for the curved distance without a chord, use the Circle Arc Calculator. If it asks for the full distance around a circle, use the Circle Circumference Calculator.
For a complete circular surface, use the Circle Area Calculator. For a wedge, circular segment, or annular ring, use the Sector, Segment and Ring Calculator. For a circle defined by coordinates or an equation, use the Circle Equation Calculator or the Circle from 3 Points Calculator.
Final verification workflow
- Confirm that What do you know? matches the two measurements actually supplied.
- Read the generated field labels exactly. In particular, distinguish Chord length, Sagitta / chord height, Radius, Diameter, and Central angle (°).
- Enter positive numbers only, keep both lengths in the same Unit, and enter the angle in degrees rather than radians.
- Check the geometry before calculating: a chord must not be longer than its diameter, and a sagitta must be smaller than its radius.
- Click Click to calculate chord and confirm that the output labels include Chord length, Radius, Diameter, Central angle, Sagitta / height, and Minor arc length.
- Compare the diameter with 2r and compare the minor arc with the fraction θ/360 of the full circumference 2πr. The curved arc should not be confused with the straight chord.
- Use Click to show steps to inspect the substitution and Click to visualize it to check that O is the centre and OM is perpendicular to AB.
- Only after checking the full result should you click Click to copy result. Use Clear before a new problem, and verify the original mode and unit when restoring an item from Recent calculations.
Open the Circle Chord Calculator to calculate a chord from the measurements you know, then return to this guide whenever you need to explain the exact controls, formulas, units, outputs, or verification process.
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