Central Angle Calculator — Arc Length to Degrees & Radians
Find the central angle of a circle — in both degrees and radians — from the radius and arc length. Also get the sector area, chord length, and the fraction of the full circumference the arc spans.
units
units
In radians: 1.5708 rad • θ = s / r
- 1
Central angle in radians: θ = s ÷ r
7.854 ÷ 5 = 1.5708 - 2
Convert to degrees: θ × (180 ÷ π)
1.5708 × (180 ÷ π) = 90.0002
How does this calculator work?
Divide arc length by radius to get the central angle in radians (θ = s/r), then multiply by 180/π for degrees. Sector area = ½r²θ; chord length = 2r sin(θ/2). Arc must not exceed the full circumference 2πr (360°). Example: arc 7.854 on radius 5 → θ ≈ 1.571 rad ≈ 90°.
Formula
How this is calculated
A central angle is an angle whose vertex is at the center of a circle, with the two sides (radii) meeting the circumference at two endpoints. The arc between those endpoints has a length s proportional to the angle: θ (in radians) = s / r. Since a full circle spans 2π radians (360°), this fraction scales linearly — a half-circle arc equals π radians or 180°. To convert radians to degrees, multiply by 180/π.
The sector is the "pie slice" cut out by the two radii and the arc. Its area equals half the square of the radius times the angle in radians: A = ½r²θ. This can also be written as A = (θ/360) × πr² when θ is in degrees. The chord is the straight line connecting the two arc endpoints; its length is 2r × sin(θ/2), derived from the isosceles triangle formed by the two radii and the chord.
All calculations assume a perfect circle and use exact formulae. The arc length must not exceed the full circumference 2πr, which would imply a central angle greater than 360°.
Frequently asked questions
A central angle is an angle formed at the center of a circle by two radii. Its measure in radians equals the arc length divided by the radius (θ = s/r). A full circle = 2π radians = 360°.
Multiply the radian value by 180/π. For example, π/2 radians × (180/π) = 90°. This calculator shows both automatically once you enter the radius and arc length.
A central angle has its vertex at the center of the circle. An inscribed angle has its vertex on the circumference. The inscribed angle theorem states that an inscribed angle is exactly half the central angle that subtends the same arc.
Also known as
TG we-Calculate Editorial Team. (2026). Central Angle Calculator — Arc Length to Degrees & Radians [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/central-angle-calculator
TG we-Calculate Editorial Team. "Central Angle Calculator — Arc Length to Degrees & Radians." TG we-Calculate. 2026. https://we-calculate.com/calculator/central-angle-calculator.
TG we-Calculate Editorial Team, "Central Angle Calculator — Arc Length to Degrees & Radians," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/central-angle-calculator
@misc{wecalculate_central_angle_calculator, title = {Central Angle Calculator — Arc Length to Degrees & Radians}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/central-angle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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