The Circle Arc Calculator is designed to solve a circular arc from whichever two related measurements you already know. It can work from a radius and central angle, a diameter and central angle, an arc length and radius, an arc length and central angle, a chord length and radius, or a chord length and central angle.
This guide specifically explains this calculator because the calculator page is intentionally short and easy to scan. The page supplies the live controls and result panel; this guide explains the actual labels, the six input modes, the formulas, a complete worked example, the diagram, the device history, and ways to verify the answer before copying it.
What is an arc length?
Arc length, written as s, is the distance measured along a curved portion of a circle between two endpoints. It is not the straight line between those endpoints. The calculator returns arc length in the selected linear unit: mm, cm, m, in, or ft.
If the central angle is α degrees and the radius is r, the calculator uses the fraction α/360 of the complete circumference:
s = 2πr(α/360)
A 90° arc therefore represents one quarter of the full circular distance, while a 180° arc represents one half. The result remains a length, not a square measurement.
What is the central angle?
The central angle is the angle formed at the circle's centre by the two radii leading to the ends of the arc. On this calculator, the input label is exactly Central angle (°), so enter degrees in that field. The result also shows the same angle in radians, but radians are an output here rather than a separate input choice.
The allowed central-angle range is greater than 0° and no greater than 360°. A full 360° input describes a complete circle. In the chord-length-and-central-angle mode, the calculator requires an angle less than 360° because a chord is being used to identify two distinct endpoints.
What are radius and diameter?
The radius, r, is the centre-to-edge distance. The diameter, d, passes through the centre from one side of the circle to the other, so d = 2r. The calculator converts a diameter to a radius before finding the arc and reports both Radius and Diameter in the selected unit.
These are linear measurements. If the selected unit is cm, for example, the radius, diameter, arc length, chord length, and sagitta are shown in cm. Only the sector-area result uses a square unit, such as cm2.
What are sector area, chord length, and sagitta?
Sector area, written as A, is the wedge-shaped area enclosed by two radii and the corresponding arc. This calculator reports it in the square of the selected unit. Its formula is:
A = πr2(α/360)
Chord length, written as c, is the straight-line distance from one endpoint of the arc to the other. It is shorter than the corresponding arc for an ordinary partial circle. The calculator uses:
c = 2r sin(α/2)
Sagitta, written as h, is the radial depth between the chord and the arc according to the calculator's displayed geometry. The tool computes it with:
h = r − √(r2 − (c/2)2)
The result panel shows all three of these related quantities: Sector area, Chord length, and Sagitta. For a major arc, interpret the displayed sagitta using the formula and diagram rather than assuming it is the height of the entire major region.
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 does not terminate. The calculator uses the built-in mathematical value of π rather than asking you to type it.
In the arc formula, 2πr is the complete circumference and α/360 selects the required fraction of that circumference. The same fraction appears in the sector-area formula because a sector is the same fraction of the full circle's area πr2. The calculator also converts the degree angle to radians internally for its trigonometric chord calculation and displays that radian value alongside the degree result.
Which “What do you know?” option should you select?
The calculator's first menu is labeled exactly What do you know?. It has six options. Choosing one changes the two input labels and placeholders, so do not assume that the first field always means radius.
| Exact option | Input labels created by the calculator | What the tool derives | Important check |
|---|---|---|---|
| Radius and central angle | Radius; Central angle (°) | Arc length, diameter, sector area, chord length, sagitta, and radians | Both values must be greater than zero. |
| Diameter and central angle | Diameter; Central angle (°) | Radius first, then the same arc outputs | The calculator divides the diameter by 2. |
| Arc length and radius | Arc length; Radius | Central angle, then the remaining outputs | The recovered angle must be no greater than 360°. |
| Arc length and central angle | Arc length; Central angle (°) | Radius, then diameter, area, chord, and sagitta | The entered angle is in degrees. |
| Chord length and radius | Chord length; Radius | Central angle and the arc-related outputs | The chord cannot be longer than the diameter. |
| Chord length and central angle | Chord length; Central angle (°) | Radius, then diameter, arc, area, and sagitta | Use a central angle less than 360°. |
How to use the Circle Arc Calculator
Open the Circle Arc Calculator and follow the controls as they appear:
- Choose one of the six choices under What do you know?.
- Read the two labels that appear and enter one positive number in each corresponding field. The placeholders begin with Enter, such as Enter radius or Enter central angle (°).
- Choose a unit under Unit. The available choices are mm, cm, m, in, and ft. The menu initially shows cm.
- Click the exact button labeled Click to calculate.
- Read the result panel and check the linear and square units before using the supporting controls.
The calculator accepts positive numeric values only. If either value is missing, zero, negative, or not a number, it reports Enter two values greater than zero. It does not provide a separate degree-versus-radian selector: the central-angle input is explicitly in degrees.
Worked example using every input mode
Use a 5 cm radius and a 90° central angle as the reference case. This is a quarter-circle, so the expected arc length is one quarter of the circumference. The exact controls and values below exercise every option in the What do you know? menu while describing the same geometry.
| What do you know? | First field | Second field | Unit | Expected central angle |
|---|---|---|---|---|
| Radius and central angle | Radius = 5 | Central angle (°) = 90 | cm | 90° |
| Diameter and central angle | Diameter = 10 | Central angle (°) = 90 | cm | 90° |
| Arc length and radius | Arc length = 7.853981634 | Radius = 5 | cm | 90° |
| Arc length and central angle | Arc length = 7.853981634 | Central angle (°) = 90 | cm | 90° |
| Chord length and radius | Chord length = 7.071067812 | Radius = 5 | cm | 90° |
| Chord length and central angle | Chord length = 7.071067812 | Central angle (°) = 90 | cm | 90° |
For each row, select the mode, enter the two numbers under the exact generated labels, select cm under Unit, and click Click to calculate. The calculator formats displayed numbers to at most four decimal places by removing unnecessary trailing zeroes. The common result is approximately:
| Output shown by the calculator | Formula or conversion | 90° quarter-circle result |
|---|---|---|
| Arc length | s = 2πr(90/360) | 7.854 cm |
| Radius | Known directly or recovered from the selected pair | 5 cm |
| Diameter | d = 2r = 2 × 5 | 10 cm |
| Central angle | Input or recovered angle | 90° |
| Radian value in parentheses | 90π/180 | 1.5708 radians |
| Sector area | A = π × 52 × (90/360) | 19.635 cm2 |
| Chord length | c = 2 × 5 × sin(90/2) | 7.0711 cm |
| Sagitta | h = 5 − √(52 − (7.0711/2)2) | 1.4645 cm |
Small differences in the last displayed digit can occur when the example's long arc or chord input is rounded before entry. The first two modes are the simplest verification because the radius and angle are entered directly.
How the reverse input modes work
Diameter and central angle
When Diameter and central angle is selected, the calculator treats the first value as d, computes r = d/2, and then applies the arc, sector, chord, and sagitta formulas. For d = 10 cm and α = 90°, the internal radius is 5 cm, so the output matches the reference example.
Arc length and radius
When Arc length and radius is selected, the calculator solves the arc formula for the angle:
α = s × 360/(2πr)
It then converts that angle to radians for the trigonometric chord calculation. A recovered angle outside the permitted 0° to 360° range produces an error rather than a result.
Arc length and central angle
When Arc length and central angle is selected, the calculator solves for radius:
r = s/[2π(α/360)]
For s = 7.853981634 cm and α = 90°, this gives r = 5 cm. The tool then derives diameter, sector area, chord length, and sagitta.
Chord length and radius
When Chord length and radius is selected, the calculator finds the central angle from:
α = 2 arcsin(c/(2r))
The input is rejected if the chord is longer than the diameter, because the check c > 2r would describe an impossible chord for that radius. For the example, c = 7.071067812 cm and r = 5 cm produce a 90° angle.
Chord length and central angle
When Chord length and central angle is selected, the calculator solves for radius:
r = c/[2 sin(απ/360)]
The degree angle is converted to radians inside the sine calculation. The calculator specifically rejects an angle of 360° or more in this mode with the instruction to use a central angle less than 360° for a chord.
How to use visualization, steps, copy, clear, and history
These controls are present on the live page and become useful after a valid calculation:
- Click to visualize it opens a diagram showing the circle, highlighted arc AB, radii from the centre O to A and B, chord AB, the central angle, the calculated arc length, and the sagitta. The page states that the diagram is a visual aid, so use the numeric result for exact measurements.
- Click to show steps reveals the working. The displayed lines include the arc-length substitution, the sector-area substitution, the chord formula, and the sagitta formula. The button changes to Hide steps while the working is open.
- Click to copy result copies a short result containing arc length, radius, and angle. After a successful copy its label briefly changes to Copied. Check the full result panel first because the copy action is not a substitute for reviewing sector area, chord, sagitta, diameter, and units.
- Clear empties the current input fields, removes the result and working, hides the diagram, and disables the visualization and copy buttons. Selecting a different mode also resets the current calculation.
- Click to view history opens the Recent calculations panel. The entries are stored on the current device, and the panel includes Clear history. Selecting a history entry restores its mode, two input values, and unit, then recalculates it. The calculator keeps up to ten recent entries and removes duplicate value/mode/unit combinations.
Where Do We Use Arc Length, Sector Area, Chord Length, Sagitta, Radius, Diameter, and Central Angle in Real Life?
The outputs answer different physical or design questions, even though they come from the same two-dimensional circle. Choosing the correct output matters more than simply obtaining a decimal number.
| Calculator output | Distinct real-life use |
|---|---|
| Arc length | Estimate the curved travel path of a wheel rim over part of a rotation, the length of a curved rail, or the material needed along a rounded edge. |
| Sector area | Estimate the surface of a fan-shaped floor section, a radar sweep region, a wedge-shaped sign, or paint and paving inside a circular slice. |
| Chord length | Set the straight span between two points on a circular panel, arch, window, tank wall, or manufactured part. |
| Sagitta | Check the rise of an arc above its chord when fabricating an arch, curved plate, roadway profile, roof member, or inspection template. |
| Radius | Use the centre-to-edge size to set a compass, locate a curve, position a circular feature, or check the clearance around a rotating part. |
| Diameter | Check the full width of a pipe opening, wheel, lid, circular cutout, or panel against the available space. |
| Central angle | Specify how much of a circular turn, sweep, sector, camera field, or curved layout is occupied by the arc. |
| Radians | Pass the angular result into trigonometry, physics, robotics, or software calculations that use radian measure even though the calculator accepts degrees. |
When this calculator is not the right tool
This tool is for a circular arc and its related chord, sector, and radial quantities. If the required result is the distance around a complete circle, use the Circle Circumference Calculator. If you need the full enclosed surface rather than a sector, use the Circle Area Calculator.
Use the Circle Chord Calculator when the primary question is the straight chord rather than the curved arc. Use the Sector, Segment and Ring Calculator when you need to distinguish a sector, a circular segment, or an annular ring. For coordinate geometry, use the Circle Equation Calculator or the Circle from 3 Points Calculator instead.
Final verification workflow
- Confirm that What do you know? matches the two measurements actually supplied by the problem.
- Check the generated labels: Radius, Diameter, Arc length, Chord length, and Central angle (°) must not be mixed up.
- Enter positive numbers in one consistent unit and remember that the angle field is in degrees, not radians.
- Click Click to calculate and check that the output includes the expected unit, square unit for sector area, and a sensible angle.
- Compare the arc with the complete circumference 2πr: for an angle α, the arc should be the fraction α/360 of that circumference.
- Use the quarter-circle example or the displayed steps to check the formula, and use Click to visualize it if the relationship between arc, chord, radii, and sagitta is unclear.
- Only after reviewing the full result should you use Click to copy result. Use Clear before a new problem, and label any history entry with its original mode and unit.
Open the Circle Arc Calculator to calculate an arc, then return to this guide whenever you need to explain the inputs, formulas, units, or verification steps.
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