Central Angle Calculator - Free Online Tool

Calculate your central angle with our free online tool.

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Angle Calculator

Central Angle Calculator

Calculate the central angle from arc length and radius

The central angle (in radians) equals the arc length divided by the radius: θ = s / r. This is the fundamental relationship between arc length, radius, and angle.

Central Angle

28.65°

28° 38' 52.4" | 0.5 rad

Sector area:25
Arc fraction:7.96%

Formula

θ = s / r = 5 / 10 = 0.5 rad = 28.65°

How This Tool Works

Calculating a central angle relies on the relationship between arc length and the radius of a circle. A central angle is the angle formed at the center point by two radii that connect to points on the circumference.

Enter the arc length and radius in matching units. The calculator divides arc length by radius to get the central angle in radians, then converts that result to degrees.

Formula: θ = s / r radians. To convert radians to degrees, multiply by 180 / π.

Why This Matters

Central angles are used in geometry, engineering design, navigation, surveying, architecture, and rotational motion analysis. They help describe what portion of a circle an arc or sector covers.

  • Navigation and surveying: Measure bearing changes and angles between mapped points.
  • Physics and mechanics: Quantify rotation in wheels, gears, and circular motion.
  • Architecture: Lay out radial designs such as domes, spokes, curves, and circular plans.

Common Mistakes to Avoid

The most common errors come from mixing units or confusing central angles with other circle angles.

  • Use matching units: If radius is measured in meters, arc length must also be in meters.
  • Do not confuse angle types: A central angle starts at the circle center. An inscribed angle starts on the circumference.
  • Check full rotations: Results above 360° may represent more than one revolution.

Tips for Best Results

Sketch the circle first and label the radius, center point, and arc. A quick diagram makes it easier to confirm that the known values match the calculator inputs.

  • Test simple cases such as quarter circles and half circles to confirm the setup.
  • Use the radian value when plugging the result into most geometry or physics formulas.
  • Use the degree value when communicating the angle for drafting, navigation, or general measurement.

Frequently Asked Questions

Common questions about the Central Angle Calculator - Free Online Tool

Enter your source units value in the input field, and the converter will automatically calculate the equivalent target units value. You can also click the swap button to reverse the conversion direction.
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Sources & References

International System of Units (SI): plane angle

Plane angle is measured in the radian (rad); 1° = π/180 rad. Conversions between SI and other units use exact, internationally agreed factors maintained by NIST.
→Source: NIST — International System of Units (SI)

International System of Units (SI)

Authoritative definitions for plane angle, from the BIPM SI Brochure (9th edition), the defining reference for the SI.
→Source: BIPM — SI Brochure (9th ed.)

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