Arc Length and Sector Area: worked example
| Radius | 10 |
|---|---|
| Central angle (degrees) | 90 |
| Arc length | 15.7 |
| Sector area | 78.5 |
| Chord length | 14.1 |
Arc Length and Sector Area: formula
Enter a radius and a central angle on the selected degree basis. The chord is the straight endpoint-to-endpoint distance, while the arc follows the circular boundary. A diameter must first be halved.
Use the same radius unit for both output lengths.
Arc Length and Sector Area: changing Radius
| Radius | Arc length | Sector area | Chord length |
|---|---|---|---|
| 8 | 12.6 | 50.3 | 11.3 |
| 9 | 14.1 | 63.6 | 12.7 |
| 10 (example) | 15.7 | 78.5 | 14.1 |
| 11 | 17.3 | 95 | 15.6 |
| 12 | 18.8 | 113.1 | 17 |
Arc Length and Sector Area: compare input ranges
| Input | Tested from | To | Arc length: low input | Arc length: high input | Swing |
|---|---|---|---|---|---|
| Central angle (degrees) | 81 | 99 | 14.1 | 17.3 | 3.1 (20%) |
| Radius | 9 | 11 | 14.1 | 17.3 | 3.1 (20%) |
Arc Length and Sector Area: individual input comparisons
| Input | Value | Arc length | Sector area | Chord length |
|---|---|---|---|---|
| Radius | 8 | 12.6 | 50.3 | 11.3 |
| 9 | 14.1 | 63.6 | 12.7 | |
| 10 | 15.7 | 78.5 | 14.1 | |
| 11 | 17.3 | 95 | 15.6 | |
| 12 | 18.8 | 113.1 | 17 | |
| Central angle (degrees) | 72 | 12.6 | 62.8 | 11.8 |
| 81 | 14.1 | 70.7 | 13 | |
| 90 | 15.7 | 78.5 | 14.1 | |
| 99 | 17.3 | 86.4 | 15.2 | |
| 108 | 18.8 | 94.2 | 16.2 |
Arc Length and Sector Area: combined input changes
| Radius / Central angle (degrees) | 72 | 90 | 108 |
|---|---|---|---|
| 8 | 10.1 | 12.6 | 15.1 |
| 9 | 11.3 | 14.1 | 17 |
| 10 | 12.6 | 15.7 | 18.8 |
| 11 | 13.8 | 17.3 | 20.7 |
| 12 | 15.1 | 18.8 | 22.6 |
Arc Length and Sector Area: which assumptions move the result?
Arc Length and Sector Area: values and definitions
| Quantity and definition | Value used |
|---|---|
| Radius | 10 |
| Central angle (degrees) — The angle at the circle's center, from 0 to 360 degrees. | 90 |
| Arc length | 15.7 |
| Sector area | 78.5 |
| Chord length | 14.1 |
Arc Length and Sector Area: half and double Central angle (degrees)
| Central angle (degrees) | Arc length |
|---|---|
| 45 | 7.9 |
| 90 | 15.7 |
| 180 | 31.4 |
Common error: Using the chord where the curve is meant. On a wide angle the two differ by a lot, and a material order sized to the chord comes up short.
Angles in radians and degrees are easily mixed up: the formulas need radians, but the calculator accepts degrees. The arc length depends on the curved path, not the chord. A segment, the area between the chord and the arc, is a different quantity.
Very small angles make the arc and chord nearly equal.
Geometry calculators: Ellipse Area Calculator, Pyramid Volume Calculator, Right Triangle & Pythagorean Calculator, Aquarium Volume Calculator, Annulus Area Calculator, Regular Polygon Calculator.
A slice of a circle is a fixed fraction of the whole, set by its angle. Ninety degrees is a quarter, so its arc is a quarter of the circumference and its sector a quarter of the area. The chord is different: it is the straight line across the arc, always shorter than the curve it cuts.
For a radius of 10 and a quarter turn, the arc runs 15.7 and the chord 14.1.
Decide which length you need. A curved edge to trim is the arc. A straight cut or beam across the opening is the chord.
See the circle area calculator for the full circle, the annulus for rings, and the ellipse page for ovals. The trigonometry calculator converts angles, the triangle side page finds chords through the law of cosines, and the regular polygon calculator handles straight-sided shapes.
What is a sector?
The region of a circle between two radii and the arc they cut, like a slice of pie.
What is the chord?
The straight line joining the two endpoints of the arc, always shorter than the arc itself.
How do I calculate arc length?
Multiply the radius by the central angle in radians, or by the angle in degrees times pi over 180.