Answer first
What this calculator tells you
Calculate arc length, sector area and chord length from a circle's radius and central angle. Size a curved edge, a pie slice or a paving arc from the radius and the angle it spans. Formula: Arc length = r × θ; sector area = ½r²θ; chord = 2r·sin(θ ÷ 2), with θ in radians. At the worked-example inputs, the arc length is 15.7. Holding every other input steady, moving radius from 8 to 12 moves the result from 12.6 to 18.8.
Transparent method
The formula
Size a curved edge, a pie slice or a paving arc from the radius and the angle it spans.
Worked example
Example inputs
How to interpret the result
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.
At the worked-example inputs the arc length is 15.7. It rises with central angle (degrees) and radius.
Formulas assume ideal geometric shapes. Real-world measurements carry their own error, and every input must use the same unit: mixing feet and inches silently produces a wrong answer.
Before you rely on it
What to check
Decide which length you need. A curved edge to trim is the arc. A straight cut or beam across the opening is the chord.
The common error
Where people go wrong with arc length and sector area calculator
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.
Sensitivity evidence
How radius changes the arc length
Holding every other input at the worked-example value, moving radius from 8 to 12 moves the arc length from 12.6 to 18.8: a spread of 6.3, or 40% of the worked-example result.
| Radius | Arc length | Sector area | Chord length |
|---|---|---|---|
| 8 | 12.6 | 50.3 | 11.3 |
| 9 | 14.1 | 63.6 | 12.7 |
| 10worked example | 15.7 | 78.5 | 14.1 |
| 11 | 17.3 | 95 | 15.6 |
| 12 | 18.8 | 113.1 | 17 |
Every input, tested
Which input moves the arc length most
Of the 2 inputs, central angle (degrees) moves the arc length most (3.1 across the range tested) and radius moves it least (3.1).
| Input | Tested from | To | Arc length at each end | Swing |
|---|---|---|---|---|
| Central angle (degrees) | 81 | 99 | 14.1 to 17.3 | 3.1 (20%) |
| Radius | 9 | 11 | 14.1 to 17.3 | 3.1 (20%) |
Two variables at once
Arc length by radius and central angle (degrees)
Across the grid the arc length runs from 10.1 to 22.6. Moving radius from 8 to 12 shifts it by 6.3 at the middle column, and moving central angle (degrees) from 72 to 108 shifts it by 6.3 at the middle row, so neither is the bigger lever here.
| 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 |
The highlighted cell is the worked example: 15.7.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Radius | 10 | Enter the radius used in this calculation. |
| Central angle (degrees) | 90 | The angle at the circle's center, from 0 to 360 degrees. |
| Arc length | 15.7 | |
| Sector area | 78.5 | |
| Chord length | 14.1 | |
Inputs, definitions and assumptions
Radius
Enter the radius used in this calculation. The prefilled worked-example value is 10.
Central angle (degrees)
The angle at the circle's center, from 0 to 360 degrees. The prefilled worked-example value is 90.
How to use this calculator
- 1Verify the inputs. Gather radius and central angle (degrees) from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the arc length at 15.7. Store your own version of it as Scenario A.
- 3Test one change. Start with central angle (degrees), the input with the biggest effect here: moving central angle (degrees) from 81 to 99 takes the arc length from 14.1 to 17.3, a swing of 20% of the worked-example figure.
- 4Check the extremes. At half the example central angle (degrees) (45) the arc length is 7.9; at double (180) it is 31.4.
People also ask
Frequently asked questions
How do you calculate arc length and sector area?
Arc length = r × θ; sector area = ½r²θ; chord = 2r·sin(θ ÷ 2), with θ in radians. At the worked-example inputs the arc length is 15.7.
What does the arc length and sector area result mean?
Size a curved edge, a pie slice or a paving arc from the radius and the angle it spans. At the worked-example inputs the arc length is 15.7. It rises with central angle (degrees) and radius.
How much does radius change the arc length?
Holding every other input at the worked-example value, moving radius from 8 to 12 moves the arc length from 12.6 to 18.8, a spread of 6.3.
What are the limits of this arc length and sector area calculator?
Formulas assume ideal geometric shapes. Real-world measurements carry their own error, and every input must use the same unit: mixing feet and inches silently produces a wrong answer. The tables on this page test radius only from 8 to 12; a value outside that range is not tabulated here.
Which input moves the arc length most in the arc length and sector area calculator?
Ranked by how far each moves the arc length across the range tested: central angle (degrees) (3.1, 20%) and radius (3.1, 20%).
If I double central angle (degrees) in the arc length and sector area calculator, does the arc length double?
Doubling it from 90 to 180 takes the arc length from 15.7 to 31.4, which is 2.00 times the worked-example figure. So the result scales almost exactly in proportion. Halving it to 45 gives 7.9.
How much does central angle (degrees) matter in the arc length and sector area calculator?
The worked example uses 90. Holding every other input at its worked-example value, moving central angle (degrees) from 81 to 99 takes the arc length from 14.1 to 17.3, a swing of 20% of the worked-example figure.
Which inputs change the sector area in the arc length and sector area calculator?
At the worked-example inputs it is 78.5. Radius takes it from 63.6 to 95 and central angle (degrees) takes it from 70.7 to 86.4.
Which inputs change the chord length in the arc length and sector area calculator?
At the worked-example inputs it is 14.1. Radius takes it from 12.7 to 15.6 and central angle (degrees) takes it from 13 to 15.2.
What's the difference between surface area and volume?
Surface area measures the total area of a shape's outer boundary: relevant for paint, wrapping or heat loss. Volume measures the space enclosed inside: relevant for capacity or material fill. Both grow with size, but at different rates.
Sources and evidence
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