Arc Length and Sector Area Calculator
This arc length calculator finds the length of a circular arc and the area of the sector it cuts from a circle. Enter the radius and the central angle in degrees or radians, and the arc length calculator returns the arc length, sector area, chord length, segment area, and sector perimeter in one step.
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The Arc Length and Sector Area Calculator is part of our Geometry collection. Browse more geometry to size, convert, and check your work alongside this tool.
What the arc length calculator does
An arc is a piece of a circle's edge, and a sector is the slice of the circle between that arc and the center. Both are simple fractions of the full circle: the angle over 360°. The arc length calculator converts the angle to radians, then applies the arc and sector formulas, and adds the straight chord across the arc and the segment between chord and arc.
Arc length formula and sector area formula
With θ in radians:
To convert degrees to radians, multiply by π ÷ 180. In degrees the arc length is s = 2πr × (angle ÷ 360).
Arc length calculator example
A radius of 10 and a central angle of 60°:
The chord is exactly 10, since a 60° sector forms an equilateral triangle, and the segment between chord and arc is 9.0586.
How to use the sector area calculator
Enter the radius and the central angle, and choose degrees or radians. The angle can be anything up to a full circle of 360° or 2π. Press Calculate. Units follow the radius: a radius in meters gives arc length in meters and areas in square meters.
Reading arc length in radians and degrees
In radians, the arc length is simply the radius times the angle, which is why radians are the natural unit for circles. The arc is always longer than the chord; for small angles they are nearly equal, and the gap grows as the angle widens. The segment area is the sector minus the triangle, useful for tank and channel calculations.
When to use an arc length of a circle calculator
- Cutting curved trim or edging
- Laying out a curved wall or path
- Finding the area of a pie slice or fan shape
- Geometry and trigonometry homework
- Bending pipe or flat bar to a radius
Common arc length calculator mistakes
- Using degrees in s = r θ. θ must be in radians.
- Using diameter for radius. Halve the diameter.
- Confusing chord and arc. The chord is the straight line.
- Mixing sector and segment. The segment excludes the triangle.
Tips for a chord length calculator check
- Measure the chord and rise on site, then solve for the radius.
- Keep extra decimals in the angle.
- For bends, use the center line radius.
Sector of a circle vs the whole circle
A sector is a fraction of the circle: a 90° sector is a quarter, so its area is a quarter of πr² and its arc a quarter of the circumference. The circle calculator gives the full area and circumference to divide. The triangle formed by the two radii and the chord has an area of ½ r² sin θ, which the triangle area calculator confirms, and subtracting it from the sector leaves the segment.
Arc length and sector area in real projects
A curved garden edge with a 4 m radius over 90° needs 6.28 m of edging, and the bed inside covers 12.57 m². When the curve is part of a regular shape, such as an octagon with rounded corners, combine this tool with the regular polygon calculator. For bending bar or pipe, measure the radius to the center line of the material, not the inside edge.
Central angle calculator table for a radius of 1
Multiply arc length by r and sector area by r²:
| Angle | Radians | Arc length | Sector area |
|---|---|---|---|
| 30° | 0.5236 | 0.5236 | 0.2618 |
| 45° | 0.7854 | 0.7854 | 0.3927 |
| 60° | 1.0472 | 1.0472 | 0.5236 |
| 90° | 1.5708 | 1.5708 | 0.7854 |
| 120° | 2.0944 | 2.0944 | 1.0472 |
| 180° | 3.1416 | 3.1416 | 1.5708 |
| 270° | 4.7124 | 4.7124 | 2.3562 |
| 360° | 6.2832 | 6.2832 | 3.1416 |
Arc Length and Sector Area Calculator FAQ
How do you find arc length?
Multiply the radius by the angle in radians: s = r θ. A radius of 10 at 60° gives 10.472.
What is the area of a sector?
A = ½ r² θ in radians, or πr² × angle ÷ 360 in degrees. A 60° sector of radius 10 has an area of 52.36.
How do I find the chord length?
c = 2 r sin(θ ÷ 2). For radius 10 at 60°, the chord is 10.
How do I find the central angle from arc length?
Divide the arc length by the radius to get radians: θ = s ÷ r.