Central Angle Calculator
Enter any two of arc length, radius and central angle, in degrees or radians, and this calculator solves for the third, then reports the full sector family around it: sector area, chord length, circular segment area, sagitta, sector perimeter, circumference and full circle area.
That solve-in-any-direction design matters because real problems rarely arrive in the textbook order. A fabricator knows the arc and the radius and needs the angle. A student knows the angle and the arc and needs the radius. A tool that only runs forward from radius and angle makes both of them rearrange formulas by hand.
Enter any two of the three values and leave the third blank. Length units are yours: enter feet and every length and area comes back in feet and square feet. If you fill in all three, the arc length and radius take priority and the angle is recomputed from them.
Quick Answer
Key takeaways
- One relationship runs everything: arc length = radius x angle in radians. Every other quantity on this page is built from it.
- Radians are not a complication, they are the shortcut. Convert degrees to radians first (x pi / 180) and every formula on the page becomes a single multiplication.
- Sector and segment are different regions with confusingly similar names. Sector includes the center; segment is the cap between chord and arc.
- At 60° the chord equals the radius. It is the fastest sanity check in circle geometry: if your chord at 60° is not the radius, an input is wrong.
How to use the central angle calculator
Fill in any two of the three fields and leave the third blank. Radius plus angle is the familiar forward problem and gives the arc. Arc plus radius gives the angle. Arc plus angle gives the radius. If you fill in all three fields, the tool treats arc length and radius as authoritative and recomputes the angle from them, so an inconsistent angle entry will be replaced rather than averaged.
Set the angle unit before entering the angle: degrees for most school and workshop work, radians for calculus and physics. The preset buttons drop in 30°, 45°, 60°, 90° or 180° and switch the unit to degrees for you. If you are entering an angle from a protractor reading in degrees, minutes and seconds, convert it to decimal degrees first with the DMS calculator.
Every length comes back in the unit you entered, and areas in that unit squared. If the arc itself is your whole problem, the dedicated arc length calculator goes deeper on that single quantity; if your shape is exactly half a circle, the area of a semi-circle calculator handles the 180° special case directly.
The three formulas that run everything
| Known | Find | Formula |
|---|---|---|
| Arc length s, radius r | Central angle | theta = s / r (radians) |
| Angle, radius r | Arc length s | s = r x theta (theta in radians) |
| Angle, arc length s | Radius r | r = s / theta (theta in radians) |
The reason radians make the master formula s = r theta exact, rather than approximate, is that the radian is defined by it: one radian is the angle for which the arc equals the radius. Degrees are a human convention, 360 of them to a turn for historical reasons, so every degree formula carries the pi / 180 conversion as baggage. Compute in radians, present in whichever unit the reader wants.
Worked example, checked against this exact tool
The calculator is pre-filled with radius 5 and angle 60°, arc blank. These are its exact outputs, to four decimal places.
| Quantity | Formula at r = 5, theta = pi/3 | Result |
|---|---|---|
| Central angle | 60° in radians | 1.0472 rad |
| Arc length | 5 x 1.0472 | 5.2360 |
| Sector area | 0.5 x 25 x 1.0472 | 13.0900 |
| Chord length | 2 x 5 x sin(30°) | 5.0000 |
| Segment area | 0.5 x 25 x (1.0472 – sin 60°) | 2.2647 |
| Sagitta | 5 – 5 x cos(30°) | 0.6699 |
| Sector perimeter | 2 x 5 + 5.2360 | 15.2360 |
| Circumference / full area | 2 pi r / pi r² | 31.4159 / 78.5398 |
Two checks are built into this example. The chord at 60° comes out at exactly the radius, 5.0000, because the two radii and the chord form an equilateral triangle at that angle. And the sector area, 13.0900, is exactly one sixth of the full circle area 78.5398, because 60° is one sixth of a turn. If either check fails on your own inputs, the angle unit is the first suspect: entering 60 with radians selected describes an angle of over 3,400 degrees, and the tool will faithfully compute that instead.
Sector vs segment: the mix-up that costs marks
Sector
- Bounded by two radii and the arc: the whole pizza slice
- Area = 0.5 r² theta
- Perimeter = 2r + arc length
Segment
- Bounded by the chord and the arc: the slice with the center cut out
- Area = sector minus the central triangle = 0.5 r² (theta – sin theta)
- The region meant by “area cut off by a chord”
Exam questions exploit the similarity of the names. If a problem mentions the center of the circle, two radii, or a slice, it wants the sector. If it mentions a chord, a line cutting the circle, or the region between a line and the curve, it wants the segment. This calculator reports both on every run, labeled, so the remaining job is simply reading the right row.
What competing calculators miss
Most central-angle tools solve one direction: radius and angle in, arc length out, and nothing else. The stronger rivals each add part of the picture. One derives the angle from chord and radius as well and shows a step-by-step arc walkthrough, but offers no quick way to enter the standard angles that dominate homework. Another solves for radius or angle from sector area, a genuinely useful mode, but reports no segment or sagitta, which are the quantities layout and arch work actually need.
This page combines the pieces in one tool: solve any two of arc, radius and angle, the complete output family (sector area, chord, segment, sagitta, sector perimeter, circumference, full circle area), common-angle presets, and degrees and radians handled symmetrically with both shown in every result. One honest limit: it solves from arc, radius and angle only. If your knowns are the chord or the sector area, you will need to rearrange by hand or use a tool built around that input; the formulas box above gives you the relationships to do it.
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Frequently Asked Questions
A central angle is the angle formed at the center of a circle by two radii. It measures the wedge of the circle cut by those radii and the arc between them. Because the vertex sits at the center, the angle and the arc are locked together by the radius: double the angle and you double the arc.
Divide the arc length by the radius. That gives the angle in radians, because radians are defined so that arc length equals radius times angle. Multiply by 180 / pi to convert to degrees. An arc of 5.236 on a radius of 5 gives 1.0472 radians, which is 60 degrees.
Multiply the radius by the angle in radians: s = r x theta. In degrees, convert first by multiplying by pi / 180. A 60 degree angle is pi/3 radians, so on a radius of 5 the arc is 5 x pi/3, about 5.236. If you would rather start from a known arc and work outward, a dedicated arc length tool runs the same relationship from the other side.
Chord = 2 x radius x sin(theta / 2), with theta in radians. The chord is the straight line between the two ends of the arc. Two landmark cases are worth remembering: at 60 degrees the chord exactly equals the radius, and at 180 degrees it equals the diameter.
A sector is the pie-slice region bounded by two radii and the arc. A segment is the smaller region bounded by the chord and the arc, with the triangular middle removed. Segment area equals sector area minus the central triangle, which is why its formula, one half r squared times (theta minus sin theta), is the sector formula with the triangle term subtracted.
The sagitta is the height of the arc above its chord, measured at the midpoint: r minus r cos(theta / 2). It is the number you need when laying out an arch, a curved wall or a lens, where the rise of the curve over a known span is the dimension that has to be built or checked.
Yes. The formulas keep working past 180 degrees, describing the major arc and the larger sector on the other side of the chord. Below 180 the sector is the smaller slice; above it, the sector is most of the circle. Nothing in the calculator caps the angle at 180, so layout problems with reflex angles are handled the same way.
Multiply degrees by pi / 180. The values worth memorizing are 30 degrees = pi/6, 45 = pi/4, 60 = pi/3, 90 = pi/2, 180 = pi and 360 = 2pi. This calculator converts automatically: pick the unit, enter the angle in it, and the result always shows both degrees and radians.
References
- OpenStax, Algebra and Trigonometry, “Angles”: radian measure defined as arc length over radius, the arc length formula s = r theta, and sector area as one half theta r squared. OpenStax: Angles. Accessed October 2026.
- LibreTexts, Precalculus (OpenStax), “Angles”: degree and radian conversion, arc length and sector area in both unit systems. LibreTexts: Angles. Accessed October 2026.
- Mathwords, “Segment of a Circle”: the segment area formula A = (r squared / 2) (theta – sin theta) with theta in radians, the basis of the segment output. Mathwords: Segment of a Circle. Accessed October 2026.
- The Math Doctors, “How to Find Any Part of a Segment of a Circle”: the seven related quantities of a circular segment (arc, chord, radius, sagitta, angle, apothem, area) and how any two determine the rest. The Math Doctors: Segment of a Circle. Accessed October 2026.