Calculator Features
📐Six practical solve modes
🧮Step-by-step working
📏Degrees and radians
🍕Major/minor sector comparison
🚫Detects impossible input combinations
🔄Length & area unit conversion
🖨Printable / PDF results
📊CSV export
🖼SVG diagram export
📋Copy results & formula
🎚Decimal precision control
📱Mobile friendly
🧭 Jump to a section
What Is a Sector?
A sector is the "pie slice" region of a circle bounded by two radii and the arc between them — think of a single slice cut from a pizza or pie. Every sector is defined by just two independent measurements: the circle's radius (r) and the central angle (θ) between the two radii.
💡 Did You Know? A sector has exactly the same two-parameter structure as an ellipse's semi-axes — which is why this calculator, like the ellipse calculator, offers several different starting combinations that all fully determine the shape, rather than forcing you into one rigid input layout.
Parts of a Circle
Radius
The distance from the circle's centre to any point on its edge — every sector calculation starts from this value.
Diameter
Twice the radius (d = 2r); the full width of the circle through its centre.
Arc
The curved portion of the circle's circumference that bounds the sector.
Chord
The straight line segment connecting the two endpoints of the arc — always shorter than the arc itself, since a straight line is the shortest path between two points.
Central Angle
The angle at the circle's centre between the two radii bounding the sector, measured in degrees or radians.
Sector Area Formula
A = (θ/360) × πr² (degrees) | A = ½r²θ (radians)
Both versions say the same thing: the sector's area is whatever fraction of the full circle's area (πr²) the angle θ represents. A sector with radius 10 and a 60° angle covers 60/360 = 1/6 of the full circle, giving an area of (1/6)×π×100 ≈ 52.36 square units.
Arc Length Formula
L = (θ/360) × 2πr (degrees) | L = rθ (radians)
The same fraction-of-the-whole logic applies to arc length: it's whatever fraction of the full circumference (2πr) the angle represents. The radians version, L = rθ, is often more convenient — it's actually the definition of a radian: the angle that makes the arc length equal to the radius.
Sector Perimeter
P = 2r + L
A sector's boundary is made of two straight radii plus the curved arc — so its perimeter is simply twice the radius, plus the arc length. Don't confuse this with the chord: the perimeter uses the curved arc, not the straight-line distance between the arc's endpoints.
Major vs Minor Sector
Any two radii divide a circle into two sectors: the minor sector (the smaller one, with angle ≤ 180°) and the major sector (the larger one, with angle > 180°), together making up the whole circle (360°). This calculator always shows both, since a question that specifies "the sector" without saying which one can be genuinely ambiguous for angles other than exactly 180°.
Beyond that broad split, this calculator classifies every sector into one of six angle bands for a quicker visual read: 🟢 Acute Sector (below 90°), 🔵 Quarter Circle (exactly 90°), 🟡 Obtuse Sector (90°–180°), 🟠 Semicircle (exactly 180°), 🔴 Major Sector (above 180°), or 🟣 Full Circle (360°).
Bounding Box
For CAD, architecture and material-cutting purposes, this calculator also reports the sector's bounding box — the smallest square that contains the full parent circle, with width and height both equal to 2r. A sector narrower than a semicircle can technically fit inside a smaller rectangle, but the full 2r × 2r square is what matters when the sector will be cut from square or round stock material.
Degrees vs Radians
Degrees split a full circle into 360 equal parts (an arbitrary but traditional choice); radians measure angle by arc length directly, with a full circle equal to 2π radians (≈ 6.2832). The formulas differ only in whether θ needs the extra θ/360 scaling — L = rθ works directly in radians, while the degrees version, L = (θ/360)×2πr, is doing that conversion internally. This calculator lets you enter angles in either unit.
Worked Examples
Pizza slice — r=15 cm, θ=45°
Area = (45/360)×π×15² ≈ 88.357 cm², Arc length ≈ 11.781 cm
Ferris wheel gondola spacing — r=20 m, θ=30°
Arc between adjacent gondolas ≈ 10.472 m, sector area ≈ 104.720 m²
Roundabout road segment — r=8 m, θ=90°
Area ≈ 50.265 m², Perimeter ≈ 28.566 m
Circular garden bed — r=6 m, θ=120°
Area ≈ 37.699 m² of planting space, chord (straight edge) ≈ 10.392 m
Stadium seating block — r=50 m, θ=40°
Area ≈ 872.665 m² of seating, arc length ≈ 34.907 m
Water tank gauge — r=4 m, arc=13.96 m
θ = L/r ≈ 3.491 rad ≈ 200° — the filled sector represents more than half the tank
Chord + arc length — c=10, L=10.472
Solved numerically: θ ≈ 60°, r ≈ 10 — exactly matching the pizza-slice-style example above
Impossible-input example
Chord entered as 20, arc entered as 15: a straight chord can never be longer than the curved arc between the same two points
The calculator correctly rejects this rather than returning a meaningless angle
Real-World Applications
| Field | Use |
|---|---|
| Architecture | Curved facades, fan-shaped windows, amphitheatre seating layouts |
| Civil engineering | Road curves, roundabouts, pipe bend sections |
| Construction | Curved garden beds, driveway aprons, arched openings |
| Mechanical engineering | Gear tooth spacing, cam profiles, sector gauges |
| Manufacturing | Sheet-metal sector blanks, pie-shaped component cutting |
| Sports facilities | Stadium seating bowls, running track curves |
| Graphic design | Pie chart segments, radial layouts, icon design |
| Education | Teaching fractions, angles and circle geometry |
Real-World Examples
🍕
Pizza & cake slices
Everyday sector shapes
Everyday sector shapes
📊
Pie charts
Data visualisation segments
Data visualisation segments
🎡
Ferris wheels
Gondola arc spacing
Gondola arc spacing
🔄
Roundabouts
Curved road segments
Curved road segments
🏟
Stadium seating
Curved seating blocks
Curved seating blocks
⚙️
Mechanical gears
Gear tooth sector spacing
Gear tooth sector spacing
Glossary
- Radius (r)
- The distance from the circle's centre to its edge.
- Central angle (θ)
- The angle at the centre between the sector's two radii.
- Arc length (L)
- The length of the curved edge of the sector.
- Chord
- The straight line connecting the arc's two endpoints.
- Sagitta (segment height)
- The perpendicular distance from the midpoint of the arc to the midpoint of the chord.
- Minor sector
- The smaller of the two sectors formed by a pair of radii (angle ≤ 180°).
- Major sector
- The larger of the two sectors formed by a pair of radii (angle > 180°).
- Radian
- The angle at which the arc length equals the radius; a full circle is 2π radians.
Common Mistakes
| Mistake | Fix |
|---|---|
| Confusing chord length with arc length | The chord is a straight line; the arc follows the curve — the arc is always the longer of the two |
| Mixing degrees and radians in one formula | Convert consistently: multiply degrees by π/180 to get radians, or divide radians by π/180 for degrees |
| Using the diameter instead of the radius | All sector formulas here use the radius; halve the diameter first if that's what you have |
| Forgetting there are two sectors for any given angle | The "other" sector (360° − θ) is the major sector, unless θ is exactly 180° |
| Assuming perimeter uses the chord | Sector perimeter is 2r + arc length, not 2r + chord — the arc is the curved boundary |
Formula Cheat Sheet
Quick Reference
Area:A = ½r²θ (rad) = (θ/360)×πr² (deg)Arc length:
L = rθ (rad) = (θ/360)×2πr (deg)Perimeter:
P = 2r + L | Chord: c = 2r×sin(θ/2) | Sagitta: h = r(1−cos(θ/2))Angle from arc:
θ = L/r | Angle from area: θ = 2A/r² | Angle from chord: θ = 2·arcsin(c/2r)Full circle = 360° = 2π rad | Major + minor sector always sum to the full circle
Practice Questions
Beginner (with answers)
- Find the area of a sector with radius 10 and central angle 90°.
- Find the arc length of a sector with radius 6 and central angle 60°.
- A sector has central angle 200°. Is it a major or minor sector?
- Find the perimeter of a sector with radius 5 and arc length 6.
- What percentage of a full circle is a 90° sector?
Show answers
1) (90/360)×π×100≈78.540 2) (60/360)×2π×6≈6.283 3) Major (angle > 180°) 4) 2×5+6=16 5) 90/360×100=25%
Advanced (with answers)
- A sector has radius 8 and area 30. Find the central angle in degrees.
- A sector has arc length 12 and central angle 40°. Find the radius.
- Find the chord length of a sector with radius 10, central angle 120°.
- A sector has chord 8 and radius 6. Find the central angle.
- Find the sagitta of a sector with radius 12, central angle 90°.
Show answers
1) θ=2×30/64≈0.9375 rad≈53.71° 2) r=12/(40×π/180)≈17.19 3) c=2×10×sin(60°)≈17.321 4) θ=2×arcsin(8/12)≈83.62° 5) h=12×(1−cos45°)≈3.515
🔑 Key Takeaways
- A sector is defined by just two values — radius and central angle — and everything else follows from them.
- Area A = ½r²θ (radians) or (θ/360)×πr² (degrees); arc length L = rθ or (θ/360)×2πr.
- Perimeter uses the arc length, not the chord: P = 2r + L.
- Any angle splits a circle into a minor sector (≤180°) and a major sector (>180°), which always sum to 360°.
- Chord + arc length has no algebraic solution — this calculator solves that combination numerically and says so.
Frequently Asked Questions
The "pie slice" region of a circle bounded by two radii and the arc between them — like a single slice cut from a pizza.
A = (θ/360)×πr² in degrees, or A = ½r²θ in radians. A sector with radius 10 and a 60° angle has an area of about 52.360 square units.
L = (θ/360)×2πr in degrees, or L = rθ in radians. A 60° sector with radius 10 has an arc length of about 10.472 units.
P = 2r + L — twice the radius, plus the arc length. This uses the curved arc, not the straight-line chord.
From the arc length: θ = L/r. From the area: θ = 2A/r². From the chord: θ = 2×arcsin(c/2r). All give the answer in radians; convert to degrees by multiplying by 180/π.
From the arc length and angle: r = L/θ (θ in radians). From the chord and angle: r = c / (2×sin(θ/2)).
A = ½r²θ (θ in radians), or equivalently (θ/360)×πr² (θ in degrees) — the fraction of the full circle's area that the angle represents.
L = rθ (θ in radians), or (θ/360)×2πr (θ in degrees) — the fraction of the full circumference that the angle represents.
Arc length follows the curve of the circle; chord length is the straight-line distance between the same two endpoints. The chord is always shorter than the arc, for any angle greater than 0°.
The larger of the two sectors formed by a pair of radii, with a central angle greater than 180°.
The smaller of the two sectors formed by a pair of radii, with a central angle of 180° or less.
Yes — every pair of radii splits a circle into exactly two sectors whose angles, areas, and arc lengths always sum to the full circle's total (360°, πr², and 2πr respectively).
Degrees split a full circle into 360 arbitrary units; radians measure angle by arc length directly, with a full circle equal to 2π radians (about 6.2832). Multiply degrees by π/180 to convert to radians, or radians by 180/π for degrees.
Because a radian is defined so that arc length equals radius times angle directly (L=rθ) with no extra scaling factor — the degrees version needs the θ/360 fraction to achieve the same thing.
Also called the segment height, it's the perpendicular distance from the midpoint of the arc to the midpoint of the chord: h = r(1−cos(θ/2)).
Yes, though it requires solving numerically rather than with a simple formula, since sin(θ/2)/θ has no elementary inverse. This calculator handles that calculation transparently and shows the working.
The relationship between chord, arc and angle involves sin(θ/2)/θ, a function that cannot be algebraically inverted — the same type of limitation that makes an ellipse's perimeter require numerical methods rather than a simple formula.
No — a straight line is always the shortest path between two points, so a chord is always strictly shorter than the arc connecting the same two points on a circle.
The diameter (2r), which occurs when the central angle is exactly 180° — at that point the "chord" is a full diameter, and the sector is a semicircle.
(θ/360)×100 for degrees, or (θ/2π)×100 for radians — simply the central angle's share of the full 360° (or 2π radian) circle.
A special sector with a central angle of exactly 180° — at that point the minor and major sectors are equal, and the "chord" becomes the full diameter.
A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc (the region between the chord and the arc, without the two radii). The sagitta this calculator reports is a key segment measurement.
A sector is a wedge-shaped piece of a single circle; an annulus is the ring-shaped region between two concentric circles of different radii — entirely different shapes, despite both being circle-related.
The distance from the circle's centre to any point on its arc — the same radius as the circle the sector is cut from.
Twice the radius (d = 2r) — the full width of the parent circle through its centre.
The circumference (2πr) is the full boundary of the parent circle; the sector's arc length is just the portion of that circumference the sector's angle covers.
Any linear unit for the radius, arc length, chord and perimeter (mm, cm, m, ft); area uses the squared version of the same unit (m², ft²); the central angle is in degrees or radians, with no length unit.
Yes — choose any of the six modes above (radius+angle, radius+arc, radius+area, arc+angle, chord+radius, or chord+arc), enter the known values, and the calculator solves everything else instantly with the working shown.
Five of the six modes are solved exactly, to floating-point precision. The chord + arc length mode uses a numerical method (bisection) accurate to many more decimal places than any physical measurement could achieve.
Yes — enter the known values for whichever mode matches, and the calculator shows the formula and full working, useful for checking homework as well as producing an answer.
Confusing chord length with arc length; mixing degrees and radians in one calculation; using the diameter where the radius is needed; and forgetting that a sector's perimeter uses the arc, not the chord.
Curved facades, fan-shaped windows and amphitheatre-style seating layouts are often designed using sector geometry.
Road curves, roundabouts and pipe bend sections are commonly analysed as circular sectors, with arc length giving the curved distance directly.
Gear tooth spacing and cam profiles often rely on sector angle and arc length calculations to space components evenly around a circle.
Sheet-metal sector blanks and pie-shaped components are cut using sector area and chord measurements to minimise waste.
Pie chart segments are sectors, with each segment's angle proportional to the data value it represents.
The calculator uses the required values for the selected mode to solve the shape, and cross-checks any extra values you entered (perimeter or area) against the result, flagging a warning if they don't match.
Yes — use Copy Results to copy everything to the clipboard, Copy Formula for just the formulas used, Export CSV for a spreadsheet-ready file, Download SVG Diagram for a vector image, or Print / Save as PDF for a printable worksheet.
A sector only has two independent measurements, so any two independent, compatible values determine it fully. Six different natural combinations are offered because different situations start with different known values — sometimes a radius and angle, sometimes just a chord and an arc.
The calculator rejects it — an arc length can never exceed the full circumference of its own circle, since the arc is only ever a portion of that circumference.
The calculator rejects it — a sector's area can never exceed the area of its full parent circle (πr²), since a sector is always a portion of that circle.
Yes — that describes the full circle itself, which this calculator recognises and labels as a special case rather than an ordinary sector.
No — a sector needs a positive central angle greater than 0° and no more than 360°; this calculator validates that range and explains why if it's violated.
No — an angle beyond 360° would simply wrap back around the same circle, so this calculator caps valid input at 360° and explains the limit if exceeded.
There's no fixed minimum size — any positive radius and any central angle between 0° and 360° describe a valid sector. The specific mode you're using may add its own constraints, like a chord never exceeding the diameter.
"Wedge" is an informal, everyday name for the same shape a mathematician calls a sector — a pizza slice, pie slice, or pie-chart segment are all sectors.
They're a quick visual read of the central angle band: Acute Sector is below 90°, Quarter Circle is exactly 90°, Obtuse Sector is 90°–180°, Semicircle is exactly 180°, Major Sector is above 180°, and Full Circle is 360°. They complement the exact angle value rather than replacing it.
The smallest square that contains the sector's full parent circle, with both width and height equal to 2r (the diameter). This is useful for working out material size when cutting a sector from square or round stock, even though a narrow sector could technically fit a smaller rectangle.
Calculation Assumptions
ℹ️ What this calculator assumes
- Results are calculated using Euclidean (flat-plane) geometry.
- Measurements are assumed to lie in a flat plane, not on a curved or sloped surface.
- Rounding to the selected decimal precision may cause very small differences between displayed values and hand calculations.
- All input values are assumed to use the same unit — mixing units (e.g. a radius in metres and a chord in centimetres) will produce an incorrect result.
- The chord + arc length mode is solved by numerical bisection, since no elementary algebraic formula exists for that combination — the result is accurate to many more decimal places than any practical measurement.
About This Calculator
Printable Formula Sheet
A one-page reference with every sector formula on this page.
Sector Formula Sheet
MegaCalcOnline.com · Area, arc length, perimeter and chord formulas
Scan for the live calculator
| Find | Formula |
|---|---|
| Area (degrees) | A = (θ/360)×πr² |
| Area (radians) | A = ½r²θ |
| Arc length (degrees) | L = (θ/360)×2πr |
| Arc length (radians) | L = rθ |
| Perimeter | P = 2r + L |
| Chord length | c = 2r×sin(θ/2) |
| Sagitta (segment height) | h = r(1−cos(θ/2)) |
| Angle from arc | θ = L/r |
| Angle from area | θ = 2A/r² |
| Angle from chord | θ = 2·arcsin(c/2r) |
References
- Khan Academy — Arcs, sectors and circle geometry
- OpenStax Mathematics — radians and circular measurement
- Maths Is Fun — Circle Sector and Segment
- NIST — SI units for length and area
- Wolfram MathWorld — Circular Sector
Every formula and worked example on this page was independently verified — see “About This Calculator” above for the full review notes.