Radius, arc, angle — give any two and get the third.
Leave one blank
Radius, arc and angle: any two give the third.
solved for the arc
10.472
From s = rθ with θ in radians: 10.472 = 10 × 1.0472. In degrees the same angle is 60°.
Central angle
60°
1.0472 radians
Arc length
10.472
along the curve
Radius
10
centre to edge
Chord length
10
straight across, not along
Sector area
52.3599
the whole wedge
Segment area
9.0586
between chord and arc
The shaded wedge is 16.67% of the circle, so its arc is that share of the circumference and its area that share of the area. The chord is the straight line closing the wedge; the segment is what sits between that line and the curve.
The arc length formula s = rθ only works with θ in RADIANS. In degrees it needs the conversion factor π/180, and forgetting it is the commonest error here — it makes the answer about 57 times too large.
This angle covers 16.67% of the circle, so the arc and the sector area are that fraction of the circumference and of the area.
The chord is the straight line between the arc’s endpoints, and the segment is the region between the chord and the arc — smaller than the sector by the triangle.
Arc, chord, sector and segment all come from the same angle in radians, so they cannot disagree with each other.
What this tool shows
s = rθ is one equation with three unknowns, so any two give the third. The catch is that θ must be in radians: using degrees makes the arc about 57 times too long, and it is the error this page is built to prevent.
Central angle from radius and arc length
Arc length from radius and angle
Radius from arc and angle
The chord across the arc
Sector area and segment area
What fraction of the circle the angle covers
Solves for any of the three The sector drawn Chord and segment too Radians handled for you
Updated 7 September 2026 · Works in any browser, no installation
The arc is the radius times the angle in radians: s = rθ. A radius of 10 through 60° gives an arc of 10 × π/3 ≈ 10.47. In degrees the same multiplication gives 600, which is wrong by a factor of 180/π — about 57.
At a glance
Formula shown
Arc length s = r\u03b8 with \u03b8 in radians. Sector area = \u00bdr\u00b2\u03b8. Chord = 2r\u00b7sin(\u03b8/2). Segment area = \u00bdr\u00b2(\u03b8 \u2212 sin \u03b8), which correctly becomes larger than the sector past 180\u00b0 because sin \u03b8 turns negative there.
Scenario support
Working out how much edging a curved path needs; finding the angle a known arc subtends; sizing a pie-chart wedge or a curved component.
Educational estimate
Planning support from the values you enter — not professional advice.
The radians trap
s = rθ is only true when θ is measured in radians. Substituting degrees is the single commonest mistake in this topic, and it is a big one: the answer comes out about 57 times too large.
A radius of 10 with a central angle of 60°: in radians that is π/3 ≈ 1.047, so the arc is about 10.47. Put 60 in instead and you get 600 — longer than the whole circumference, which is only about 62.8.
A quick sanity check catches it every time: the arc can never exceed the circumference unless the angle exceeds a full turn. If your arc is longer than 2πr for an angle under 360°, you used degrees.
This page takes degrees and converts internally, so the trap does not arise here. It is worth knowing for when you are doing it by hand.
Arc, chord and segment
Four quantities come out of the same angle, and they are easy to confuse.
Arc is the curved distance along the circle. It is rθ.
Chord is the straight line between the arc’s two endpoints. It is 2r·sin(θ/2), and it is always shorter than the arc — the straight route beats the curved one.
Sector is the whole wedge, from the centre out. Its area is ½r²θ.
Segment is just the part between the chord and the arc — the sector with its triangle removed. Its area is ½r²(θ − sin θ).
Cutting a pie gives you a sector. Cutting straight across the crust gives you a segment.
A radian, defined properly
A radian is the angle at which the arc is exactly as long as the radius. That is the definition, and it is why s = rθ is so simple.
Since the full circumference is 2πr, a full turn is 2π radians. So 360° = 2π rad, and one radian is 180/π ≈ 57.29578°.
Because it is a ratio of two lengths, a radian is dimensionless. That is why radians can be multiplied by a radius to give a length, and degrees cannot — a degree is an arbitrary subdivision with no such relationship.
It is also why calculus uses radians throughout. The derivative of sin x is cos x only in radians; in degrees it picks up a factor of π/180 and every formula gets uglier.
Where it gets used
Curved construction. Edging for a curved path, a bent handrail, a length of flexible ducting round a bend. The arc is what you order.
Machining and gears. Tooth spacing, pitch circles and rotation angles are all arc-and-angle problems.
Pie charts. A category’s share of the whole, times 360°, is its central angle. Reading one backwards is this calculation.
Navigation. Distance along a great circle is a central angle at the Earth’s centre times the Earth’s radius, which is exactly s = rθ.
Anything rotating. A wheel turning through an angle rolls its rim length forwards, which is why rθ also describes distance travelled.
Past 180°, the segment grows
Below 180° the segment is smaller than the sector, since the triangle between the chord and the centre has been removed.
Past 180° it is larger. The chord now cuts off a triangle on the far side of the centre, so that triangle is added rather than subtracted — you are looking at the major segment.
The formula ½r²(θ − sin θ) handles the switch on its own, because sin θ goes negative above 180° and subtracting a negative adds. That is neat rather than coincidental: the same expression is correct on both sides, which is a good reason to trust it.
It is worth knowing because an assertion that “the segment is always smaller than the sector” is false, and a page that enforced it would be wrong half the time.
Degrees of latitude
One degree of latitude is about 111 kilometres, and this formula is where that comes from.
Earth’s mean radius is about 6,371 km. One degree is π/180 radians, so the arc is 6371 × 0.01745 ≈ 111.2 km. A minute of arc is about 1.85 km — which is where the nautical mile came from, and why it is defined as it is.
Longitude behaves differently: the circles of longitude shrink towards the poles, so a degree of longitude is 111 km at the equator and nothing at all at the pole. Latitude is constant because its circles are great circles.
Sources and methodology
The definitions are standard, and the radian is an SI unit with a formal definition; these are the references.
Method. Whichever box is left blank is solved for, from the same relation rather than from three separate formulas, so the three arrangements cannot disagree. Chord, sector and segment are all derived from the one angle in radians for the same reason. The suite checks, across 1,080 combinations of angle and radius, that the arc is the same fraction of the circumference as the angle is of a turn and the sector the same fraction of the area — and that the segment follows the sector-minus-triangle rule below 180° and the sector-plus-triangle rule above it. That engine is verified on every change against 82 hand-written assertions, including that arc and sector are the correct fraction of the circle across 1,080 combinations, and that one radian comes out as 57.29577951° to eight decimal places. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Coterminal AngleThe principal angle between 0° and 360°, the nearest negative one, and the family θ + 360n they all belong to — reduced by exact division, not by looping.
Reference AngleThe acute angle to the x-axis — never the y-axis — with the quadrant rule written out using your own numbers and the sign of each trig function beside it.
Angle Between Two VectorsDot product, magnitudes and the angle as separate steps, in the plane or in space — with exactly parallel vectors returning 0° rather than a floating-point smudge.
Complementary and Supplementary AnglesBoth at once, plus the explement — and an honest answer when there is none, since an obtuse angle has no complement and a negative number is not one.
Clock AngleThe angle between the hands with the hour hand where it really is — at 3:30 that is 75°, not the 90° most people answer — and the face drawn to show which angle it is.
Square RootThe exact square root first — 72 gives 6 root 2 — then the decimal to as many as sixty places, computed on whole numbers rather than a double.
If the angle needs reducing into one turn first — a rotation that has gone round more than once — the Coterminal Angle Calculator does that exactly.
Educational use disclaimer
This is an educational tool. Arc and area involve π, so those results are computed in double precision and shown to a stated number of places rather than as exact values.
Published the central angle page solving for whichever of radius, arc and angle is left blank, since all three arrangements get asked and one formula covers them.
The radians trap is named outright: s = rθ needs radians, and substituting degrees makes the arc about fifty-seven times too long.
The segment formula is left to handle angles past 180° on its own — sin θ turns negative there, so the triangle is added rather than subtracted, and the segment is correctly LARGER than the sector.
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