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CalcMax

Sector Area Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 ° – 360 °

Result

13.0900 cm²

Area

Arc length
5.2360 cm
Chord
5.0000 cm

A sector calculator takes the radius of a circle and a central angle, and returns three measurements of the wedge those two cut out: the area, the arc length and the chord. A sector is the shape of a slice of pie — the region bounded by two radii and the arc between them — and it is the first page in this group where the form asks for two things of different kinds. The radius is a length and the angle is an angle, and neither can be worked out from the other: a wide slice of a small circle and a narrow slice of a large one are different shapes, so both numbers are genuinely needed. That is worth noticing because the circle page next door is the opposite — give it any one of four measurements and it produces the rest, because a circle is fixed by a single number. A sector is a circle with a bite taken out of it, and how big the bite is has to be said separately. The three outputs answer three different questions about the same wedge. The area is how much surface it covers, which is what you want for a slice of pizza, a plot of land on a curved boundary, or the opening of a camera shutter. The arc length is how far it is along the curved edge, which is what you want for a piece of trim bent around a curve or a length of fencing on a curved boundary. The chord is the straight line closing the wedge off, which is what you want for the flat edge of a slice or the width of the opening it subtends. All three come out of the same pair of numbers, and all three are a fixed fraction of the whole circle — the angle over 360 — applied to the corresponding measurement of that circle. Each formula is worth knowing on its own, and the three are set out below.

Common sectors, from a sixth of a circle to all of it

Radius (cm)Central angle (°)Area (cm²)Arc length (cm)Chord (cm)
56013.095.2365
59019.6357.8547.0711
518039.269915.70810
536078.539831.41590
109078.539815.70814.1421
612037.699112.566410.3923
12270339.29256.548716.9706
50000

Eight sectors, and the first two columns are the two inputs — which is worth stating, because every other table in this group has its inputs hidden in the shape rather than printed. The first row is what the page loads with, and it is the one to read first: at 60 degrees the chord is exactly 5, the same as the radius, because the sine of 30 degrees is exactly a half. The fourth row is the whole circle, and its area and arc length are exactly what the circle page prints for a radius of 5 — 78.5398 and 31.4159 — while its chord is 0, because the two ends have met. The fifth row is a different circle at a different angle with the same area as the fourth: a 10 cm radius at 90 degrees covers 78.5398 square centimetres, the same as a 5 cm circle covers entirely, which is a reminder that the area depends on both inputs. The seventh row is past the half way mark, where the chord is measuring the short way across a slice that is three quarters of the circle; the page computes it without complaint, and reading which of the two regions a chord has cut off is the segment page's job rather than this one's. The last row is an angle of zero, where the wedge has collapsed onto a single radius and all three readings are zero. Every number here is recomputed from its radius and angle when the page is built, in centimetres and degrees, and the four decimals are the same four decimals the results panel uses.

Formula

A = (θ ÷ 360) × πr² L = (θ ÷ 360) × 2πr c = 2r × sin(θ ÷ 2)

Radius
The distance from the centre to the curved edge, in centimetres. Half the wedge's straight edges, and the one number that says how big the circle it was cut from is
Central angle
The angle between the two straight edges, measured at the centre, in degrees. With the radius it fixes the wedge completely, and it is the number that says what fraction of the circle you have taken — 90 is a quarter of it, 60 is a sixth
Area
The surface the wedge covers, in square centimetres. It works out as the fraction of the circle times the whole circle's area, and it is the only one of the three that goes with the square of the radius: double the radius and the area goes up four times
Arc length
The distance along the curved edge, in centimetres. Same fraction of the circle, applied to the circumference instead of the area — so it goes with the radius rather than its square, and it comes to exactly the radius when the angle is one radian
Chord
The straight line from one end of the arc to the other, in centimetres. It is the flat side of the slice, and the only one of the three outputs that needs trigonometry rather than just a fraction: it is twice the radius times the sine of half the angle
sin(θ ÷ 2)
The sine of half the angle, and half rather than the whole angle is the part worth holding on to — the chord and the two radii make an isosceles triangle, and dropping a line down its middle splits the angle in two as well as the chord
θ ÷ 360
The fraction of the whole circle the wedge takes up. It is the factor the area and the arc length share, and it is why both of them are simply a scaled-down copy of the circle's own measurements
Four decimal places
How wide every reading is written. The area and the arc length carry a π and so almost never end; the chord is exact only at the angles where the sine is, which is why 60 degrees and 180 degrees come out as clean numbers and most other angles do not

The three outputs answer three different questions about the same slice, and picking the right one is most of the work. Area is for anything bought or counted by surface: a slice of pizza priced by the tray, a wedge of land between two fences meeting at a corner, a pane of glass cut to a curved frame, or the fraction of a camera's field of view a particular shutter setting lets through. Arc length is for anything that follows the curve: the length of trim needed around a curved counter, the fencing along a boundary that bulges, the belt travel around part of a pulley, or the distance a point on the rim moves when the wheel turns through a given angle. The chord is for the straight line — the width of a fan blade at its tip, the flat back of a slice that sits against a plate, the span of an arch, or how far apart two points on a circle end up after both are rotated by the same angle from a common centre. And there is a fourth use that runs through all three: the sector is the unit that a great many circular things are built out of — a pie chart is nothing but sectors, a hexagon is six of them, and a gear tooth is one with the sharp end trimmed.

Worked examples

  1. A 60 degree slice of a 5 cm circle

    1. Fraction of the circle: 60 ÷ 360 = 1/6
    2. Area: (1/6) × π × 5² = 25π ÷ 6 = 13.0899…, which rounds to 13.09
    3. Arc length: (1/6) × 2π × 5 = 10π ÷ 6 = 5.2359…, which rounds to 5.236
    4. Chord: 2 × 5 × sin(30°) = 10 × 0.5 = 5

    The input the page loads with, and the row that shows what the fraction is doing: a 60 degree slice is exactly a sixth of the circle, so its area is a sixth of 25π and its arc is a sixth of 10π. The chord is the part worth pausing on — it comes out at exactly 5, the same as the radius, because the sine of 30 degrees is exactly one half and the 2 and the half cancel. Six of these wedges laid point to point make a regular hexagon, and that is the next page over.

  2. A quarter of a 5 cm circle

    1. Fraction of the circle: 90 ÷ 360 = 1/4
    2. Area: 25π ÷ 4 = 19.6349…, which rounds to 19.635
    3. Arc length: 10π ÷ 4 = 7.8539…, which rounds to 7.854
    4. Chord: 2 × 5 × sin(45°) = 10 × 0.7071068 = 7.0710678…, which rounds to 7.0711

    The quarter circle. The chord of a 90 degree sector is the hypotenuse of a right triangle whose two legs are both the radius, so it is r√2 — here 5√2, the same 7.0711 the square page prints for its diagonal and the same number the circle page's inscribed square would need. Three pages, one number, and all three are the same triangle seen from different sides.

  3. A half circle

    1. Fraction of the circle: 180 ÷ 360 = 1/2
    2. Area: 25π ÷ 2 = 39.2699…, which rounds to 39.2699
    3. Arc length: 10π ÷ 2 = 15.7079…, which rounds to 15.708
    4. Chord: 2 × 5 × sin(90°) = 10 × 1 = 10

    The angle where the chord stops being a chord and becomes a diameter: the two ends of the arc are now opposite each other, so the straight line between them passes through the centre and is 2r. The sine of 90 degrees is exactly 1, which is why the answer is clean. Everything on this row is half of the full circle below it except the chord, which is at its largest here — past 180 degrees the chord starts shrinking again even as the area keeps growing.

  4. A third of a 6 cm circle

    1. Fraction of the circle: 120 ÷ 360 = 1/3
    2. Area: 36π ÷ 3 = 12π = 37.6991…, which rounds to 37.6991
    3. Arc length: 12π ÷ 3 = 4π = 12.5664…, which rounds to 12.5664
    4. Chord: 2 × 6 × sin(60°) = 12 × 0.8660254 = 10.3923048…, which rounds to 10.3923

    A third of a circle of radius 6, and the row that links this page to the next one over. The chord is 6√3, which is exactly the short diagonal of a regular hexagon of side 6 — and that is not a coincidence: three of these wedges make half a hexagon, and six of them make the whole thing. The area being 12π and the arc being 4π are the other thing to notice: when the fraction comes out tidy, the answers keep the π and drop everything else.

  5. The whole circle

    1. Fraction of the circle: 360 ÷ 360 = 1
    2. Area: 1 × π × 5² = 25π = 78.5398…, which rounds to 78.5398
    3. Arc length: 1 × 2π × 5 = 10π = 31.4159…, which rounds to 31.4159
    4. Chord: 2 × 5 × sin(180°) = 10 × 0 = 0

    A 360 degree sector is the whole circle, and the area and arc length are exactly what the circle page prints for a radius of 5 — the same 78.5398 and the same 31.4159. They have to be: a sector that takes up all of the circle is the circle. The chord is the interesting one. It is 0, and that is the correct answer rather than a failure: the two ends of the arc have come round to meet each other, so the straight line between them has no length at all. A chord can be zero while the area is at its largest — the two measurements have nothing to do with each other.

Limitations

The page is for sectors only: two straight radii and the arc between them, on a flat circle. A segment — the region between a chord and its arc, which is what you get if you cut the triangle off this wedge — is a different shape with its own page next door. The readings are in centimetres and square centimetres whatever unit the dropdown is set to, so a radius entered in inches comes back as a centimetre answer you have to convert yourself; the angle is in degrees and the dropdown can take radians or gradians, but the result is always worked out from degrees. Angles above 360 degrees are refused rather than wrapped around: once past a full turn there is more than one way to read which slice you meant, and the page would rather say so than guess. Four decimal places is a display width rather than a claim about precision — the area and the arc length carry a π and almost never end, and the chord is exact only at the angles where the sine is. The chord measures the straight line across the wedge, so for angles above 180 degrees it is measuring the short way across a slice that is more than half the circle, which is correct but is rarely what someone drawing a pie chart has in mind. Nothing here handles a slice of an ellipse or any other non-circular curve, a wedge with a rounded tip, or the thickness of a physical slice such as a cake or a wedge of cheese.

Frequently asked questions

Why does this page need two inputs when the circle page needs one?
Because a sector is not fixed by one number. A circle is: give its radius and you know everything about it. A sector is a circle with a bite taken out, and how big the bite is has to be said separately — a 30 degree slice of a large circle and a 180 degree slice of a small one are entirely different shapes. The radius fixes the circle and the angle fixes how much of it you have taken, so both are needed and neither can be worked out from the other.
Why is the chord exactly the same as the radius at 60 degrees?
Because the chord is 2r·sin(θ/2), and at 60 degrees the half angle is 30 degrees, whose sine is exactly one half. The 2 and the half cancel and you are left with r. It is one of the few angles where the chord comes out as a whole number, and it is why a hexagon works: six 60 degree wedges laid point to point give a shape whose every side — the six radii and the six chords — is the same length.
What is the difference between a sector and a segment?
A sector is the wedge bounded by two radii and an arc, like a slice of pie. A segment is the region between a chord and its arc — which is what is left of the wedge once you cut off the triangle formed by the two radii and the chord. The segment page next door computes that other region, and its area is this page's area minus the triangle.
Which of the three readings do I want?
Area if the slice is a surface — a slice of pizza, a wedge of land, a pane of glass. Arc length if something follows the curve — trim around a curved counter, fencing on a bulging boundary, the belt travel on part of a pulley. Chord if you need the straight line across the slice — the width of a fan blade at its tip, the span of an arch, or how far apart two points on the rim end up.
The chord is 0 for a full circle. Is that a bug?
No, it is the right answer. At 360 degrees the two ends of the arc have come all the way round to meet each other, so the straight line between them has no length. It is worth seeing that the area is at its largest — the whole circle — at the same moment the chord is at its smallest, because those two measurements have nothing to do with each other. Only chords have this behaviour; the arc length simply becomes the circumference.
Can I enter the angle in radians?
Yes, the dropdown next to the angle takes degrees, radians or gradians, and converts whatever you type into degrees before working anything out. The entries in the reference table are in degrees throughout, so a radian entry of about 1.047 is the same slice as the 60 degree row. Nothing in the output changes with the unit — the area is in square centimetres and the arc and the chord are in centimetres either way.

References

Related calculators

Segment Area CalculatorArea between a chord and its arc, from radius and angle.Circle CalculatorRadius, diameter, circumference and area from any one of them.Arc Length CalculatorThe length of the arc between two points on a circle.Central Angle CalculatorThe angle an arc subtends at the centre of its circle.