Circle Arc Calculator: Find Length & Angle

A circle arc calculator works out the length of an arc (a portion of a circle’s circumference) from the radius and the central angle, and it can also give you the angle from arc length and radius. You enter the radius and either the angle in degrees or radians (or the arc length), and you get the arc length, the sector area, and sometimes the chord length. The calculator below uses the standard formulas and shows the result instantly.

Formula

The arc length L is a fraction of the circumference. The full circumference is 2πr, and the fraction is the central angle divided by 360° (or 2π radians). So:

• In degrees: L = (Ï€ × r × θ) / 180
• In radians: L = r × θ

Where:

• L = arc length (same unit as radius, e.g., meters, feet, cm)
• r = radius of the circle (unit of length)
• θ = central angle (degrees or radians)

The sector area (the pie-slice shape) is A = (θ/360) × πr² in degrees, or A = (1/2)r²θ in radians. The chord length (the straight line between the two endpoints of the arc) is c = 2r × sin(θ/2) when θ is in degrees (or use the radian equivalent).

Worked Example

Using the default values: radius = 5 m, central angle = 60°.

1. Arc length: L = (π × 5 × 60) / 180 = (π × 300) / 180 = 5.235987… m, rounded to 5.24 m.
2. Sector area: A = (π × 5² × 60) / 360 = (π × 25 × 60) / 360 = 13.089969… m², rounded to 13.09 m².
3. Chord length: c = 2 × 5 × sin(30°) = 10 × 0.5 = 5.00 m.

So for a 60° arc on a 5 m radius circle, the arc length is 5.24 m, the sector area is 13.09 m², and the chord is 5 m. (Notice the chord equals the radius because 60° is a regular hexagon’s central angle.)

When to Use an Arc Calculator

You’d use this in geometry homework, woodworking, metal fabrication, or any design that involves curved edges. For example:

• Cutting a curved piece of trim for a circular window.
• Laying out a garden path that follows a circular arc.
• Calculating the length of a curved conveyor belt or cable.
• Finding how much fabric you need for a circular skirt (the waist arc).

The key is that you need to know the radius and the central angle. If you only know the chord and the arc height (sagitta), you can derive the radius first, but that’s a different calculation.

How to Read the Results

The arc length is a linear measurement, so if you input meters, the result is in meters. The sector area is in square units (m², ft²). The chord length is also linear. The central angle is independent of the circle’s size; it just tells you how much of the circle you’re covering. A 90° arc is one quarter of the circumference, regardless of radius.

Be careful with units: if you mix feet and inches, convert first. The calculator assumes the radius is in the same unit throughout.

Common Mistakes and Pitfalls

1. Using degrees in the radian formula: If you use L = rθ with θ in degrees, you’ll get a wildly wrong number. Always convert: θ_rad = θ_deg × π/180.
2. Confusing arc length with chord length: The arc is the curved part; the chord is the straight line. They are equal only when the angle is 0° (trivial).
3. Forgetting to halve the angle for chord: The chord formula uses sin(θ/2), not sin(θ).
4. Rounding too early: If you round π to 3.14, your result may be off by a couple percent. Use a calculator with π or at least 3.14159.

Limitations and Extensions

This calculator assumes a perfect circle and a central angle between 0° and 360°. If your angle is greater than 360°, you can subtract 360° to get the coterminal angle, but the arc length for a full circle is just the circumference. For an ellipse, the arc length has no elementary formula; you’d need an elliptic integral. For a spiral or other curve, this doesn’t apply.

If you have the arc length and radius but need the angle, rearrange: θ_deg = (180 × L) / (π × r). If you have the arc length and the angle, solve for radius: r = (180 × L) / (π × θ_deg).

Frequently Asked Questions

How do I find the arc length without a calculator?

Use the formula L = (π × r × θ) / 180 for degrees. Multiply the radius by π, then by the angle, then divide by 180. For example, a 45° arc on a 10 cm circle: (π × 10 × 45) / 180 = (π × 450) / 180 = 7.85 cm.

What is the difference between arc length and circumference?

Circumference is the total distance around the circle (2πr). Arc length is a portion of that, corresponding to a specific central angle. If the angle is 360°, the arc length equals the circumference.

Can I use this calculator for radians?

Yes, but you need to convert the angle to degrees first. The calculator accepts degrees only. Multiply radians by 180/π to get degrees. For example, 1 radian ≈ 57.2958°.

How do I calculate the radius from arc length and angle?

Rearrange the formula: r = (180 × L) / (π × θ_deg). If you have the arc length and the central angle in degrees, plug them in. For instance, if L = 20 cm and θ = 90°, r = (180 × 20) / (π × 90) = 3600 / 282.743 = 12.73 cm.

What is a sagitta and how is it related?

The sagitta is the distance from the midpoint of the chord to the arc (the “height� of the arc). It’s used when you don’t have the radius. If you know the chord length c and the sagitta s, the radius is r = (c² + 4s²) / (8s). Then you can find the angle and arc length.

Why does the chord length formula use sin(θ/2)?

Because the chord forms an isosceles triangle with the two radii. The perpendicular from the center to the chord bisects the angle and the chord. In the right triangle, sin(θ/2) = (chord/2) / r, so chord = 2r sin(θ/2).

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in r | Radius | m | 5 | 0
in a | Central angle | deg | 60 | 0
out arclen | Arc length | m | (pi * r * a) / 180 | 2
out area | Sector area | m² | (pi * r^2 * a) / 360 | 2
out chord | Chord length | m | 2 * r * sind(a / 2) | 2