Area of a sector calculator
A sector is the slice of a circle cut by two radii, the shape of a pizza slice, and its area is the central angle’s fraction of the whole circle. Enter the radius with either the central angle (in degrees or radians) or the arc length, and this calculator returns the sector area, the arc length, the angle in both units, and the slice’s share of the full circle, with the full-circle area beside it so you can check the proportion yourself.
Results are rounded for display: area, arc and the full circle to 4 decimal places, the angle in radians to 6. The unit of the area is the square of the unit you used for the radius.
Quick Answer
A sector with radius 10 and a 60° central angle has an area of 52.3599 square units and an arc length of 10.472. That is 16.67% of the full circle, whose area is 314.1593, and the check is built in: 314.1593 ÷ 6 = 52.3599. The same slice entered the other way, as radius 10 with arc length 10.472, returns the angle (1.0472 radians) and an area of 52.36.
Key takeaways
- A sector is a fixed fraction of its circle: the angle divided by 360° (or by 2π in radians). Double the angle and you double the area; double the radius and the area quadruples.
- Radius plus angle, or radius plus arc length, are interchangeable starting points. The arc mode recovers the angle as arc ÷ radius, which is the route exam questions that hand you an arc length expect.
- The share-of-circle line is the fastest sanity check on any sector answer: 60° must come out near one-sixth of πr², or something was typed wrong.
- Use the radius, never the diameter. Because the radius is squared, that one slip multiplies the answer by four.
- Quick Answer
- How to use the sector calculator
- The sector formulas, in degrees and in radians
- Worked example: a 60 degree slice of radius 10, both ways
- Sector or segment: the boundary this calculator draws
- Angles past a full turn, and what the numbers round away
- What competing sector calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the sector calculator
Pick the pair of values you know, type them in, and click Calculate. The results stay hidden until you do.
- Choose the mode: "Radius and central angle" if you have the angle, "Radius and arc length" if you measured along the curve instead. Switching modes hides the previous results so a stale answer never sits next to new inputs.
- Enter the radius: the distance from the centre to the edge, in any unit. A 14 inch pizza has a radius of 7 inches, not 14.
- Enter the angle or the arc: for the angle, pick degrees or radians to match what you were given. The pre-filled example is radius 10 at 60 degrees; the arc field is pre-filled with 10.472, the matching arc length, so both modes describe the same slice.
- Read all five outputs: the sector area is the headline number, and the arc length, the angle in both units, the share of the circle and the full-circle area let you cross-check it from three directions before you trust it.
The tool works in whatever unit you bring: centimetres in, square centimetres out. It does not convert between units, and it reports the angle back in both degrees and radians either way, which is handy when a formula sheet expects the unit you were not given. For the neighbouring jobs this page does not do, the arc length calculator starts from the circumference side, and the central angle calculator solves for the angle itself from an arc and radius.
The sector formulas, in degrees and in radians
With the angle in radians, sector area = (θ/2) × r² and arc length = r × θ. With the angle in degrees, sector area = (θ/360) × πr² and arc length = (θ/360) × 2πr. The two pairs are the same statements: a radian measure is just the degree measure rescaled so a full turn is 2π instead of 360.
Arc length = r × θ (radians) | Angle from arc: θ = arc ÷ r | Radians = degrees × π ÷ 180
The radian forms are short because the radian is defined by the circle itself: one radian is the angle at the centre whose arc length equals the radius, a definition the SI Brochure published by the Bureau International des Poids et Mesures (BIPM) treats as the coherent unit of angle. That is also why the arc mode works at all: if arc = rθ, then θ = arc ÷ r, and the area can equally be written as ½ × arc × r. The angle and arc-length treatment in most textbooks follows the same path; OpenStax Precalculus, Section 5.1 on angles works through arc length and sector area in exactly this order, and the area result itself is the circle-area proposition of Euclid's Elements, Book XII, applied to a fraction of the disc.
Worked example: a 60 degree slice of radius 10, both ways
Radius 10 at 60 degrees is the pre-filled example, and every figure below is a real run of the calculator above.
Run it backwards to see the arc mode land in the same place: enter radius 10 and arc length 10.472, and the angle comes back as 1.0472 radians, shown as 60.0001 degrees because 10.472 is the rounded arc. The area reads 52.36 rather than 52.3599 for the same reason: you fed it a rounded input. Six more runs, covering both modes and both angle units:
| Inputs | Sector area | Arc length | Angle shown | Share |
|---|---|---|---|---|
| r = 10, angle 60° | 52.3599 | 10.472 | 60° (1.047198 rad) | 16.67% |
| r = 5, angle 90° | 19.635 | 7.854 | 90° (1.570796 rad) | 25% |
| r = 7.5, angle 1.2 rad | 33.75 | 9 | 68.7549° (1.2 rad) | 19.1% |
| r = 10, arc 10.472 | 52.36 | 10.472 | 60.0001° (1.0472 rad) | 16.67% |
| r = 0.5, angle 270° | 0.589 | 2.3562 | 270° (4.712389 rad) | 75% |
| r = 10, angle 360° | 314.1593 | 62.8319 | 360° (6.283185 rad) | 100% |
The last row is the boundary worth remembering: at 360 degrees the "slice" is the whole disc, the sector area equals πr² exactly, and the arc is the full circumference. Any angle or arc you enter below that ceiling behaves the same predictable way, scaling linearly with the angle and with the square of the radius.
Sector or segment: the boundary this calculator draws
A sector is bounded by two radii and the arc between them, and it always includes the centre of the circle. A segment is bounded by a chord and the arc, the piece you get from one straight cut, and it never includes the centre. Confusing the two is the second most common sector error after the diameter slip.
| Question | Sector | Segment |
|---|---|---|
| Bounded by | Two radii and the arc | One chord and the arc |
| Includes the centre? | Always | Never |
| Area formula | (θ/2) × r², θ in radians | Sector area minus the triangle: ½r²(θ − sin θ) |
| At 60°, r = 10 | 52.3599 | 9.0586 (52.3599 − 43.3013 of triangle) |
This calculator computes sectors only, so if your shape is a segment, run the sector first and subtract the triangle ½r² sin θ by hand, or work from the chord side with the circle formula calculator. Two boundary cases sit inside the sector family and need no special handling: at exactly 180 degrees the sector is a semicircle, the shape the area of a semi-circle calculator treats on its own, and at 90 degrees it is a quadrant, a quarter of the disc.
Angles past a full turn, and what the numbers round away
The calculator applies the formulas to any positive angle you give it, including angles past 360 degrees, and reports shares past 100% without erroring. A 450 degree entry returns an area of 392.6991 on radius 10, which is 125% of the circle.
That is honest arithmetic and a dishonest shape. A geometric sector runs from just above 0 degrees up to a full turn; between 180 and 360 degrees it is called a major sector (the badge in the results names it), and past 360 the slice would have to overlap itself. Read a result over 100% as a prompt to check the entry, not as an answer, unless you are deliberately modelling rotation, where angles past one turn are routine. The same candour applies at the small end: results display rounded to 4 decimal places (the angle in radians to 6), so a tiny sector on a small radius can show an area of 0.589 for a true 0.589049, and re-entering a rounded arc, as the worked example shows, shifts the last digits. For homework and material estimates that rounding is invisible; for chained calculations, carry the unrounded values yourself.
What competing sector calculators miss
Sector tools look interchangeable until you push on the inputs each one accepts:
- Angle-only tools (AJ Designer and similar): radius and degrees, sometimes radians, and nothing else. Given an arc length, which is how sector problems are often set, you first have to recover the angle elsewhere. This page takes the arc directly and recovers the angle for you.
- The everything-tools (OnlyCalculators, EasyUnitConverter): they add chord length and segment area, which is genuinely useful, but the sector answer gets buried among four outputs and the unit pickers multiply the ways to mis-set a field. Their worked examples also lean on chord and segment results most visitors never asked for.
- Engineering visualisers (Firgelli): a live diagram and segment area aimed at actuator and mechanism design. Strong if you are sizing hardware, heavy if you only need the area of a slice and a way to check it.
- Reference-style pages (Omni's sector and segment articles): deep explanations and special-case shortcuts for semicircles and quadrants, spread across separate sector, segment and semicircle tools, so one problem can mean three pages.
- The gap this page fills: both genuine starting points (angle in either unit, or arc length) in one tool, the angle reported back in degrees and radians together, and two built-in checks most rivals skip: the slice as a percentage of the circle and the full-circle area beside the answer, so a wrong entry shows up as an impossible proportion before it shows up as a wrong area.
Frequently asked questions
With the angle in radians, the area is (θ/2) × r². With the angle in degrees, the area is (θ/360) × πr². Both forms say the same thing: the sector covers the angle's fraction of the full circle, whose area is πr².
Arc length is r × θ with θ in radians. For a 60 degree slice of radius 10, θ is 1.0472 radians, so the arc is 10 × 1.0472 = 10.472, in the same units as the radius.
Yes, as long as you also know the radius. Choose the radius and arc length mode: the calculator recovers the angle as θ = arc ÷ r, then applies the same area formula. With radius 10 and arc 10.472, the angle is 1.0472 radians and the area is 52.36.
A sector is bounded by two radii and the arc between them, the pizza slice shape, and it always includes the centre. A segment is bounded by a chord and the arc, the piece cut off by one straight cut, and its area is the sector area minus the triangle formed by the two radii and the chord. This calculator works on sectors only.
Radians tie the angle directly to the radius: one radian is the angle whose arc length equals the radius, so arc = rθ and area = θr²/2 with no conversion factor. Degrees work too, and this calculator converts them for you, but the degree forms carry the extra ÷360 step.
The calculator accepts it and applies the same formulas, so a 450 degree entry on radius 10 returns 392.6991, which is 125% of the circle. That is arithmetic, not a shape: a real sector runs from just above 0 degrees to 360, and a slice between 180 and 360 degrees is called a major sector. Treat a share past 100% as a data entry error unless you are modelling rotation.
Yes, and it is the most common sector mistake. Every formula on this page uses the radius, half the diameter. Entering the diameter of a 14 inch pizza (14 instead of 7) quadruples the area, because the radius is squared in the formula.
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References
- Bureau International des Poids et Mesures (BIPM). The International System of Units (SI Brochure), including the definition of the radian. Accessed October 24, 2026.
- OpenStax, Rice University. Precalculus 2e, Section 5.1: Angles, arc length and sector area. Accessed October 24, 2026.
- Weisstein, E. W. Circular Sector, MathWorld: A Wolfram Web Resource. Accessed October 24, 2026.
- Encyclopaedia Britannica. Circle: mathematics, definition and properties. Accessed October 24, 2026.
- Joyce, D. E., Clark University. Euclid's Elements, Book XII: propositions on areas of circles. Accessed October 24, 2026.