The Sector, Segment and Ring Calculator is designed for partial circular regions and annular regions rather than for a complete circle. It can calculate a sector from a radius and central angle or from a radius and arc length, a minor segment from a radius and central angle or from a radius and sagitta, and a ring from radii or diameters.
This guide specifically explains this calculator because the calculator page is intentionally short and easy to scan. It identifies every mode, field, unit, result, formula, validation rule, and supporting control so that you can choose the right circular region before entering values. Use the live calculator for the numerical answer, then use this guide to understand what the displayed measurements mean and to verify the result before copying it.
What is a sector?
A sector is the portion of a circle bounded by two radii and the arc between their endpoints. The two radii meet at the circle centre and form the central angle. A sector therefore has two straight sides and one curved side.
For a radius r and a central angle θ measured in degrees, the calculator finds sector area, arc length, sector perimeter, and chord length. The sector area is only the stated fraction of the complete circle, not the area of the full circle.
What is a minor segment?
A minor segment is the smaller region cut from a circle by a chord and its corresponding arc. Unlike a sector, it does not include two radii as part of its boundary. The calculator supports a minor segment, so its central angle must be no greater than 180°.
The chord is the straight line joining the two arc endpoints. The sagitta is the perpendicular height from the midpoint of that chord to the arc. The tool reports minor segment area, chord length, sagitta, central angle, and arc length.
What is a ring?
A ring, also called an annular region, is the area between two concentric circles. The larger circle has outer radius R, and the smaller circle has inner radius r. The centre is shared, but the two radii are different.
The calculator can accept either the two radii or the two diameters. It converts diameters to radii before calculating the ring area. It also reports radial width, outer circumference, and inner circumference.
What does π mean in these calculations?
π, pronounced “pi,” is the constant ratio of a circle’s circumference to its diameter. Its decimal expansion begins 3.14159265359 and continues without ending. The calculator uses the JavaScript mathematical constant π internally and displays results rounded to four decimal places, with trailing zeroes removed.
Pi is needed because every sector, segment, and ring is based on circular geometry. The sector formulas use a fraction of a full revolution, the segment formulas use trigonometric relationships on the radius, and the ring formula subtracts one circular area from another. The angle fraction is θ/360 when θ is entered in degrees.
Which option should you select?
Start with the measurement that the problem actually gives you. Under the exact control label What do you want to calculate?, select the option that matches both the region and the known measurements.
| Exact calculator option | Use it when you know | Primary result | Important condition |
|---|---|---|---|
| Sector: radius and central angle | A radius and the angle at the centre | Sector area, arc length, sector perimeter, and chord | Central angle must be greater than 0° and less than 360° |
| Sector: radius and arc length | A radius and the curved edge length | Sector area, central angle, sector perimeter, and chord | Arc length must be less than one full circumference |
| Segment: radius and central angle | A radius and the central angle subtending a minor segment | Minor segment area, chord, sagitta, and arc length | Central angle must be no greater than 180° |
| Segment: radius and sagitta | A radius and the cap height above the chord | Minor segment area, chord, central angle, and arc length | Sagitta must not exceed the radius |
| Ring: outer and inner radius | The two centre-to-edge distances | Ring area, radial width, and both circumferences | Inner radius must be smaller than outer radius |
| Ring: outer and inner diameter | The full widths of the outer and inner circles | Ring area, radial width, and both circumferences | Inner diameter must be smaller than outer diameter |
What are the calculator fields and units?
The mode menu is labelled What do you want to calculate?. Its selection controls the two numeric fields below it. The first field is always the first measurement named in the selected option, and the second field is the second measurement. The live placeholders are Enter radius, Enter central angle (°), Enter arc length, Enter sagitta, Enter outer radius, Enter inner radius, Enter outer diameter, and Enter inner diameter, as applicable.
The unit menu is labelled Unit and offers mm, cm, m, in, and ft. The page initially shows cm as the selected unit. Enter both lengths in the same unit. Angles are entered as degrees in the field labelled Central angle (°) and are not converted into a length unit.
| Unit selected | Length outputs | Area outputs |
|---|---|---|
| mm | mm | mm2 |
| cm | cm | cm2 |
| m | m | m2 |
| in | in | in2 |
| ft | ft | ft2 |
What formulas does the calculator use?
Sector formulas
For the sector modes, the calculator uses the following relationships:
Sector area: A = πr2(θ/360).
Arc length: s = 2πr(θ/360).
Sector perimeter: P = 2r + s.
Chord length: c = 2r sin(θ/2).
When Sector: radius and arc length is selected, the calculator first recovers the angle with θ = 360s/(2πr). It then uses that angle in the same sector formulas.
Minor segment formulas
The segment area is the area of the corresponding sector minus the area of the isosceles triangle formed by the two radii and the chord:
Minor segment area: Asegment = πr2(θ/360) − (1/2)r2sin(θ).
Chord length: c = 2r sin(θ/2).
Sagitta: h = r(1 − cos(θ/2)).
Arc length: s = 2πr(θ/360).
When Segment: radius and sagitta is selected, the calculator obtains the angle from θ = 2 cos−1(1 − h/r), expressed in degrees, and then applies the segment formulas.
Ring formulas
For a ring, the outer radius is R and the inner radius is r:
Ring area: A = π(R2 − r2).
Radial width: w = R − r.
Outer circumference: Couter = 2πR.
Inner circumference: Cinner = 2πr.
For Ring: outer and inner diameter, the calculator first converts the inputs with R = D/2 and r = d/2.
How to use the Sector, Segment and Ring Calculator
- Open the Sector, Segment and Ring Calculator.
- Choose one of the six exact options under What do you want to calculate?.
- Read the two labels that appear and enter only positive numbers in the two matching fields.
- Choose mm, cm, m, in, or ft under Unit.
- Click Click to calculate.
- Review the result panel and confirm that the output is for the intended region.
Do not type a degree symbol, unit abbreviation, or formula into a numeric field. Enter the number only. Keep the radius, arc length, sagitta, and diameter values in one consistent length unit.
Worked example using every input option
The following examples use the exact option names and control labels from the live calculator. Select cm under Unit in each case. After entering the two values, click Click to calculate.
| Selected option under What do you want to calculate? | Field 1 | Field 2 | Displayed major outputs |
|---|---|---|---|
| Sector: radius and central angle | Radius = 10 | Central angle (°) = 90 | Sector area = 78.5398 cm2; arc length = 15.708 cm; sector perimeter = 35.708 cm; chord = 14.1421 cm |
| Sector: radius and arc length | Radius = 10 | Arc length = 15.7079632679 | Sector area = 78.5398 cm2; central angle = 90°; sector perimeter = 35.708 cm; chord = 14.1421 cm |
| Segment: radius and central angle | Radius = 10 | Central angle (°) = 90 | Minor segment area = 28.5398 cm2; chord = 14.1421 cm; sagitta = 2.9289 cm; arc length = 15.708 cm |
| Segment: radius and sagitta | Radius = 10 | Sagitta = 2 | Minor segment area = 16.3501 cm2; chord = 12 cm; central angle = 73.7398°; arc length = 12.87 cm |
| Ring: outer and inner radius | Outer radius = 10 | Inner radius = 5 | Ring area = 235.6194 cm2; radial width = 5 cm; outer circumference = 62.8319 cm; inner circumference = 31.4159 cm |
| Ring: outer and inner diameter | Outer diameter = 20 | Inner diameter = 10 | Ring area = 235.6194 cm2; converted outer radius = 10 cm; converted inner radius = 5 cm; radial width = 5 cm |
Checking the sector example
For Sector: radius and central angle with Radius 10, Central angle (°) 90, and Unit cm, the angle fraction is 90/360 = 1/4. Therefore:
A = π × 102 × (90/360) = 78.5398 cm2.
The arc length is s = 2π × 10 × (90/360) = 15.708 cm. The two radii add 20 cm, so the sector perimeter is 20 + 15.708 = 35.708 cm.
Checking the segment example
For Segment: radius and central angle with Radius 10 and Central angle (°) 90, the corresponding quarter-sector area is 78.5398 cm2. The triangle area is 50 cm2, so the minor segment area is 78.5398 − 50 = 28.5398 cm2.
Checking the ring example
For Ring: outer and inner radius with Outer radius 10 and Inner radius 5, subtract the inner disk area from the outer disk area:
A = π(102 − 52) = π(100 − 25) = 235.6194 cm2.
How to read the result panel
A successful sector calculation shows Sector area, Arc length, Sector perimeter, Central angle, and Chord. A successful segment calculation shows Minor segment area, Chord length, Sagitta, Central angle, and Arc length. A successful ring calculation shows Ring area, Outer radius, Inner radius, Radial width, Outer circumference, and Inner circumference.
Area results use a superscript 2, such as cm2. Arc length, chord, sagitta, perimeter, radial width, and circumference are linear measurements and use the selected unit without the square. Central angle is displayed in degrees.
How to use visualization, steps, copy, clear, and history
After a successful calculation, Click to visualize it becomes available. Click it to display a diagram. For a sector or minor segment, the diagram marks the centre as O, the arc endpoints as A and B, the chord AB, and the selected region. For a ring, it draws the outer and inner circles and identifies the outer radius, inner radius, ring area, and radial width. The page warns that the diagram is a visual aid and that the numeric result should be used for exact measurements.
Click Click to show steps to reveal the formula substitution. The button changes to Hide steps while the working is visible. The sector steps show sector area, arc length, and sector perimeter. Segment steps show sector area minus triangle area, chord, and sagitta. Ring steps show ring area and radial width.
Click Click to copy result after checking the selected mode, both inputs, unit, and output type. The copied text is a compact result summary. The control is disabled until a valid calculation has been made.
Click Clear to empty the current input fields and remove the result, steps, visualization, and copy-ready state. Changing the selection under What do you want to calculate? also redraws the correct fields and resets the current calculation.
Click Click to view history to open the Recent calculations panel. The toggle changes to Hide history. The panel stores recent calculations on the current device and offers Clear history. Each history entry can be selected to restore that calculation. Because the history label is a summary, verify the original mode, values, and unit after restoring an entry.
Where Do We Use Sector Area, Segment Area, and Ring Area in Real Life?
These outputs answer different measurement questions even when all three shapes come from a circle. Select the shape that matches the physical region instead of treating every partial circle as a sector.
| Calculator output | Distinct real-life use |
|---|---|
| Sector area | Estimate the surface covered by a fan-shaped floor design, a wedge of a circular garden, a rotating sweep, or a slice-shaped panel. |
| Arc length | Estimate the length of curved edging, rail, trim, cable, or track along only part of a circular boundary. |
| Sector perimeter | Estimate all boundary material around a wedge-shaped part, including both straight radii and the curved edge. |
| Minor segment area | Measure the cap-shaped portion cut from a circular window, tank end, arch, or curved panel by a straight chord. |
| Chord length | Determine the straight span between two points on a curved wall, circular opening, or arc-shaped piece of material. |
| Sagitta | Check the rise of an arch or the clearance between a chord-like support and a curved roof, bridge, or circular rim. |
| Ring area | Estimate the material in a washer, gasket, circular border, pipe flange, track bed, or other annular surface. |
| Radial width | Check the thickness of a circular border or the radial thickness of a washer, seal, or annular design. |
| Outer and inner circumference | Estimate the two curved edge lengths of a ring, such as the outside and inside edges of a gasket or circular track. |
| Central angle | Set out the angular opening of a fan, wedge, circular sign, or curved construction detail from a radius and arc or sagitta measurement. |
When this calculator is not the right tool
This calculator is for sectors, minor segments, and concentric rings. It is not the best choice when the problem concerns a complete circle or a different geometric relationship.
- Use the Circle Area Calculator for the area of a complete circle from a radius or diameter.
- Use the Circle Circumference Calculator for the full distance around one circle.
- Use the Circle Arc Calculator when the main task is arc length and related circular measurements.
- Use the Circle Chord Calculator when the main unknown is a chord from radius, angle, or distance information.
- Use the Basic Circle Calculator when you need the standard radius, diameter, circumference, and complete-circle area from one known circle measurement.
- Use the Circle Theorem Calculator for theorem-based relationships and diagram checks rather than direct sector, segment, or ring measurement.
Final verification workflow
- Identify the actual region: a sector has two radii, a minor segment has a chord and cap, and a ring lies between two concentric circles.
- Match the known measurements to one exact option under What do you want to calculate?.
- Check the two generated field labels before entering values. Do not substitute a diameter for a radius or an arc length for a central angle.
- Enter positive numbers only and keep both length inputs in the same selected unit.
- Check the range: a sector angle is greater than 0° and less than 360°; a minor-segment angle is no greater than 180°; a sagitta does not exceed its radius; and an inner ring measurement is smaller than its outer measurement.
- Click Click to calculate, then confirm that the result heading says Sector area, Minor segment area, or Ring area as intended.
- Check dimensions. Areas must be square units, while lengths and circumferences must be ordinary length units.
- Use Click to show steps to inspect the substitution, and use Click to visualize it to check the shape when the geometry is easy to misread.
- Only after these checks should you click Click to copy result. Use Clear before beginning a different problem.
Open the Sector, Segment and Ring Calculator for the calculation, and return to this guide whenever you need to distinguish a sector from a segment or ring, confirm a formula, or audit the displayed output.
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