Central Angle Calculator

Find the central angle of a circle from its arc length or chord length and radius, plus the resulting chord length and sector area.

Quick Facts

From arc length
theta (radians) = arc length / radius
Multiply by 180/pi to convert to degrees; a full circle is 2*pi radians (360 degrees).
From chord length
theta = 2 x arcsin(chord / (2 x radius))
Chord length can never exceed the diameter (2r).

Your Results

Calculated
Central angle
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In degrees
In radians
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theta = arc length / radius
Chord length
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Straight-line distance across the arc
Sector area
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Bounded by the two radii and the arc

Ready

Enter a radius and either an arc length or a chord length, then press Calculate.

Understanding the Central Angle

A central angle is formed at the center of a circle by two radii that extend out to two points on the circle's edge. The size of that angle determines how much of the circle's arc lies between those two points — a small central angle cuts off a short arc, while a central angle close to 360° sweeps nearly the whole circle. This calculator finds the central angle from either the arc length or the chord length, given the radius, and reports the angle in both degrees and radians along with the resulting chord length and sector area.

The core formulas

  • From arc length: θ (radians) = arc length ÷ radius — a rearrangement of s = rθ. Multiply by 180/π to convert the result to degrees.
  • From chord length: θ (radians) = 2 × arcsin(chord ÷ (2 × radius)), derived from the isosceles triangle formed by the two radii and the chord.
  • Chord length from the angle: chord = 2 × radius × sin(θ/2).
  • Sector area: area = ½ × radius² × θ (radians) — the pie-slice region bounded by the two radii and the arc.

Degrees vs. radians

Radians are the natural unit for these formulas because arc length equals radius times angle only when the angle is measured in radians. A full circle is 2π radians, or 360°; a straight line through the center (a semicircle) is π radians, or 180°. This calculator reports both units so you can use whichever your context requires.

Common applications

  • Finding the angle subtended by a chord or arc in surveying, machining, or CAD work
  • Computing the area of a circular sector — a pie slice, a rounded panel, or a pipe cut at an angle
  • Converting between arc length and angle when laying out curved paths, gears, or circular tracks

Frequently Asked Questions

What is a central angle?
A central angle is an angle whose vertex sits at the center of a circle, with its two sides passing through two points on the circle. It is directly related to the arc length between those two points: for a given radius, a larger central angle corresponds to a longer arc.
How do I find a central angle from arc length?
Divide the arc length by the radius: θ (in radians) = arc length ÷ radius. Multiply by 180/π to convert the result to degrees. For example, an arc length of 15.708 on a radius-10 circle gives θ = 1.5708 radians, or 90°.
How do I find a central angle from chord length?
Use θ = 2 × arcsin(chord ÷ (2 × radius)). This comes from splitting the isosceles triangle formed by the two radii and the chord into two right triangles. The chord length can never exceed the diameter (2r).
What is the difference between arc length and chord length?
Arc length is the curved distance along the circle between the two points; chord length is the straight-line distance connecting them. The two are close only for a small central angle — for larger angles the arc is always longer than the chord.

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