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CalcMax

Segment Area Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 ° – 360 °

Result

7.1350 cm²

Area

Chord
7.0711 cm
Arc length
7.8540 cm
Segment height
1.4645 cm

A segment calculator takes the radius of a circle and a central angle, and returns the area of the region between the chord those two cut and the arc above it, along with the chord, the arc and the height of that region. A segment is what you get if you take a sector — the pie slice bounded by two radii and an arc — and cut off the triangle formed by the two radii and the chord. What is left is the lune-shaped piece between the straight line and the curve, and it is the shape of a slice of apple, an arch window, the cross-section of a partly filled pipe, or the region a chord cuts off a circular field. The arithmetic is a subtraction once you have the two pieces: the sector's area is the fraction of the circle, and the triangle's area is half the radius squared times the sine of the angle. Take one from the other. That subtraction is worth understanding rather than just applying, because it is the point where this page stops being able to tell you which answer you wanted. A chord divides a circle into two regions — a small one and a big one — and the same radius and angle describe both of them depending on which side of the chord you mean. Below half a circle there is no ambiguity. Above it, the formula quietly crosses over to the other one.

Common segments, from a thin sliver to the whole circle

Radius (cm)Central angle (°)Area (cm²)Chord (cm)Arc length (cm)Segment height (cm)
5907.1357.07117.8541.4645
5602.264755.2360.6699
518039.26991015.7085
612022.110710.392312.56643
527071.40497.071123.56198.5355
10180157.07962031.415910
536078.5398031.415910
500000

Eight segments, and the first two columns are the inputs. Read the first two rows together to see how fast a segment shrinks: the 60 degree row has two thirds of the angle of the 90 degree row and less than a third of its area, because at small angles the triangle is eating nearly all of the wedge. The third row is 180 degrees, where the triangle has no area at all and the segment is simply the half circle — 39.2699, exactly what the sector page prints for its half circle. The fifth row is the one to read carefully: at 270 degrees the sine is negative, the subtraction turns into an addition, and the page returns 71.4049 — the larger of the two regions the chord cuts off rather than the smaller. Its chord is 7.0711, the same as the 90 degree row, because sin(135°) and sin(45°) are equal: two angles, one chord, and the segment is ten times the size. The last row is a full turn, where the chord has shrunk to 0 and the segment has grown into the entire circle, with a height of 10 — the full diameter, which is as far as an arc can ever bulge. The row before it is an angle of zero, where the segment has collapsed onto nothing at all. 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 − ½r²sin θ c = 2r sin(θ ÷ 2) h = r(1 − cos(θ ÷ 2))

Radius
The distance from the centre to the arc, in centimetres. Half the base of the triangle being cut off, and the one number that says how big the circle was
Central angle
The angle the chord subtends at the centre, in degrees. It fixes the segment along with the radius: it sets both how much of the circle the sector takes and how tall the triangle being removed is
Area
The surface of the segment, in square centimetres — the sector's area minus the triangle's. Below 180 degrees this is the smaller of the two regions a chord makes; above it the subtraction turns into an addition and you get the larger one
Chord
The straight line closing the segment off, in centimetres, and the base of the triangle that was removed. It grows to a diameter at 180 degrees and then shrinks back, reaching zero at a full turn
Arc length
The distance along the curved edge, in centimetres — the same as the sector's arc, since cutting the triangle off does not change the curve. It is the outer boundary of the segment and it simply keeps growing with the angle
Segment height
How far the arc bulges above the chord at its middle, in centimetres. It is the depth of the segment, and the quantity you would measure with a rule laid across the flat side — it runs from 0 at no angle up to the full diameter at a whole turn
½r² sin θ
The area of the triangle the two radii and the chord make. It peaks at 90 degrees and falls away to nothing at 180, which is why a half circle has no triangle to subtract
Four decimal places
How wide every reading is written. Every one of the four carries a π or a sine and so almost never ends, and the area is a difference of two such numbers, which means its last digits are the least trustworthy on the page

This page is for the region a straight line cuts off a circle, which turns out to be a more common shape than the wedge it is derived from. An arch window or a doorway with a curved top is a segment sitting on a rectangle; so is the cross-section of a tunnel whose top is an arc and whose sides are straight. Partly filled pipes and tanks are the classic case: the wetted area of a horizontal cylinder is a segment, which is why the volume of liquid in a barrel is a segment problem and not a circle one. On a map, the area between a straight property line and a curved road or river is a segment, and so is the sliver of a circular field left outside a straight fence. In manufacturing, a chord cut across a round bar or a circular blank leaves a segment as offcut, and the height formula is the one to reach for when what you can measure is how deep the cut goes rather than the angle — a rule laid across the flat and down to the deepest point of the curve gives the height directly, and the angle can be worked back from it. And it is the shape of a bite: a bite taken out of a round biscuit is a segment, with the chord across the teeth marks and the height the depth of the bite.

Worked examples

  1. A quarter-circle segment

    1. Sector area: (90 ÷ 360) × π × 5² = 19.6349…
    2. Triangle area: ½ × 5² × sin(90°) = 12.5
    3. Segment area: 19.6349 − 12.5 = 7.1349…, which rounds to 7.135
    4. Chord: 2 × 5 × sin(45°) = 7.0711
    5. Arc length: (90 ÷ 360) × 2π × 5 = 7.854
    6. Height: 5 × (1 − cos(45°)) = 5 × 0.2928932 = 1.4645

    The input the page loads with, and the clearest case of what the subtraction does. The sector is 19.6349 and the triangle removes 12.5 of it, leaving 7.135 — a third of what the wedge covered. The height makes the shape concrete: the arc rises only 1.4645 centimetres above the chord, so this is a fairly flat sliver rather than a fat piece of the circle.

  2. A shallow segment from 60 degrees

    1. Sector area: (60 ÷ 360) × π × 5² = 13.0899…
    2. Triangle area: ½ × 5² × sin(60°) = 12.5 × 0.8660254 = 10.8253
    3. Segment area: 13.0899 − 10.8253 = 2.2646…, which rounds to 2.2647
    4. Chord: 2 × 5 × sin(30°) = 5
    5. Arc length: (60 ÷ 360) × 2π × 5 = 5.236
    6. Height: 5 × (1 − cos(30°)) = 5 × 0.1339746 = 0.6699

    The same circle as the row above but a narrower angle, and the segment is much smaller than the change in angle suggests — 2.2647 against 7.135 for two thirds of the angle. The reason is that the triangle is eating almost all of the wedge: the sector is 13.0899 and the triangle takes 10.8253 of it. At small angles a segment is a thin sliver, and the area falls off much faster than the angle does.

  3. A half circle

    1. Sector area: (180 ÷ 360) × π × 5² = 39.2699…
    2. Triangle area: ½ × 5² × sin(180°) = 0
    3. Segment area: 39.2699 − 0 = 39.2699
    4. Chord: 2 × 5 × sin(90°) = 10
    5. Arc length: (180 ÷ 360) × 2π × 5 = 15.708
    6. Height: 5 × (1 − cos(90°)) = 5 × 1 = 5

    The angle where the triangle vanishes. At 180 degrees the chord is a diameter and the two radii lie flat along it, so the triangle has no area to subtract and the segment is the whole half circle — 39.2699, which is exactly what the sector page prints for its half circle. The height is 5, the radius, which is the most a segment of this circle can bulge.

  4. A third of a circle

    1. Sector area: (120 ÷ 360) × π × 6² = 37.6991
    2. Triangle area: ½ × 6² × sin(120°) = 18 × 0.8660254 = 15.5885
    3. Segment area: 37.6991 − 15.5885 = 22.1106…, which rounds to 22.1107
    4. Chord: 2 × 6 × sin(60°) = 10.3923
    5. Arc length: (120 ÷ 360) × 2π × 6 = 12.5664
    6. Height: 6 × (1 − cos(60°)) = 6 × 0.5 = 3

    The row that lines up with the pages either side of this one. The chord is 6√3 and the arc is 4π, the same numbers the sector page prints for a 120 degree wedge of the same circle — the chord and the arc do not change when the triangle is removed, because neither of them is part of it. The height is exactly 3, half the radius, because the cosine of 60 degrees is a half.

  5. The major segment at 270 degrees

    1. Sector area: (270 ÷ 360) × π × 5² = 58.9049
    2. Triangle area: ½ × 5² × sin(270°) = 12.5 × (−1) = −12.5
    3. Segment area: 58.9049 − (−12.5) = 71.4049
    4. Chord: 2 × 5 × sin(135°) = 7.0711
    5. Arc length: (270 ÷ 360) × 2π × 5 = 23.5619
    6. Height: 5 × (1 − cos(135°)) = 5 × 1.7071068 = 8.5355

    The row to read carefully, and the one that explains why this page gives no verdict on whether an answer is sensible. At 270 degrees the sine is negative, so subtracting the triangle adds 12.5 instead of taking it away, and the page returns 71.4049 — the larger of the two regions the chord cuts off, not the smaller. That is the correct answer to the arithmetic and it is very often not the region the reader had in mind, which is why nothing here is labelled as a small or a large segment. The chord is 7.0711, the same as the 90 degree row above, because sin(135°) equals sin(45°): two different angles, one chord.

Limitations

The page is for circular segments only: a straight chord and the arc above it on a flat circle. Above 180 degrees the arithmetic crosses over and returns the larger of the two regions the chord cuts off rather than the smaller — that is deliberate and unavoidable, since the formula cannot be told which one you meant, and it is why no answer here is labelled as a major or a minor segment. The area is a difference of two numbers that each carry a π or a sine, so its last digits are the least reliable on the page. The chord and the arc do not change when the triangle is removed, so a reader who wanted the wedge rather than the lune should use the sector page next door. 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. The height is measured from the chord to the deepest point of the arc and is the quantity to reach for when the angle is unknown, but the page does not work backwards from a height to an angle. Angles above 360 degrees are refused rather than wrapped around. Nothing here handles a segment of an ellipse, a chord that does not span the full arc, or the thickness and end caps of a physical piece such as a barrel stave or an arch brick.

Frequently asked questions

How is a segment different from a sector?
A sector is the wedge bounded by two radii and an arc — a slice of pie. A segment is what is left when you cut the triangle off that wedge: just the region between the chord and the arc. The sector page next door computes the wedge, and this page computes what remains after ½r²sin θ is taken away from it.
Why does the page give the bigger region at 270 degrees?
Because the triangle's area is ½r²sin θ, and past 180 degrees the sine is negative, so subtracting it adds instead of takes away. The arithmetic is right and the answer is the larger of the two regions the chord cuts off. The page deliberately does not label answers as major or minor: from a radius and an angle alone there is no way to tell which of the two regions you meant, so it computes one and leaves the reading to you.
What is the segment height for?
It is how far the arc bulges above the middle of the chord — the depth of the segment. It is on the page because it is the measurement you can actually take: lay a rule across the flat side and measure down to the deepest point of the curve and you have it, no angle needed. It runs from 0 at no angle up to the full diameter at a whole turn, and it is the usual way to describe a partly filled pipe.
Why are the chord and the arc the same as on the sector page?
Because neither of them is part of the triangle that gets cut off. The subtraction removes the region enclosed by the two radii and the chord; the chord itself is the boundary of that triangle, not part of its interior, and the arc is not touched at all. So a 120 degree segment and a 120 degree sector of the same circle have identical chords and identical arcs and different areas.
What is this shape used for?
Anything where a straight line crosses a circle. A partly filled horizontal pipe or tank has a segment as its cross-section of liquid, which is why the volume of liquid in a barrel is a segment problem. An arch window, a tunnel roof on straight walls, the area between a straight property line and a curved road, the offcut when a chord is sawn across a round bar, and a bite taken out of a round biscuit are all segments.
Can I work out the angle from the height instead?
Not on this page — the height is an output, not an input. The angle can be recovered from it, since the height is r(1 − cos(θ/2)), but that means solving for the cosine and taking an inverse, and the page does not do it. If what you have is a depth measurement, the practical route is to try angles until the height column matches what you measured, or to work the inverse out separately.

References

Related calculators

Sector Area CalculatorThe slice of a circle cut by two radii: area, arc length and chord from the 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.