Free Central Angle Calculator

Last updated October 10, 2026 · Report an error

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A central angle is an angle whose vertex sits at the center of a circle, with its two sides extending out as radii to the circle's edge. It's connected to the circle's radius and the arc length it cuts off by one of the simplest and most important relationships in geometry: s = rθ, where θ must be measured in radians. Given any two of these three values — angle, radius, or arc length — the third can always be found by rearranging this single formula. This relationship underlies an enormous range of practical work: engineers use it to size gears and pulleys from belt-wrap angles, surveyors and civil engineers use it to lay out curved roads and railway tracks from a specified radius and central angle, astronomers use a version of it to convert angular measurements of the sky into physical distances, and it is the direct ancestor of the definition of the radian itself — a radian is defined as the central angle for which the arc length exactly equals the radius.

How it works

Picture a circle of radius r. Sweep out a central angle θ starting from one radius; the length of the arc s traced out by the endpoint is directly proportional to both the radius and the angle, giving the formula s = rθ. This only works when θ is expressed in radians, because a radian is defined precisely so that an angle of 1 radian at the center always produces an arc equal in length to the radius — a convenient built-in unit conversion that degrees don't have. Choose which of the three values you want to solve for, enter the other two, and the calculator converts your angle to radians if you entered degrees, then applies s = rθ (or one of its two rearrangements, θ = s/r or r = s/θ) to find the missing value.

Enter solve for.
Enter radius (r).
Enter arc length (s).
Enter central angle (θ).
Enter angle unit.
Click Calculate to see your results.

Examples

r=10, s=15.708

θ = s/r = 15.708/10 = 1.5708 radians, which converts to 90° — a quarter of the full circle, since 15.708 is exactly a quarter of the circle's full circumference (2π×10 ≈ 62.83).

r=5, θ=60°

Converting 60° to radians gives π/3 ≈ 1.0472. Then s = rθ = 5 × 1.0472 ≈ 5.236 units of arc length.

s=12, θ=1.5 radians

r = s/θ = 12/1.5 = 8 — the radius of the circle that produces a 12-unit arc when swept through a 1.5-radian central angle.

Common mistakes to avoid

Using degrees directly in the s = rθ formula without converting to radians first, which produces an arc length far too small.
Confusing arc length (a curved distance) with the straight-line chord length between the same two points — they are not the same measurement.
Mixing up central angle with inscribed angle, which is exactly half the central angle for the same arc.
Forgetting that r, s, and θ must all use consistent length units (e.g., all in meters) — mixing centimeters and meters mid-calculation silently corrupts the result.

Frequently asked questions

Because s = rθ only holds when θ is measured in radians — that's actually the geometric definition of a radian (the angle for which arc length equals the radius). Using degrees directly in this formula gives a wrong answer roughly 57.3 times too small.

A central angle has its vertex at the circle's center; an inscribed angle has its vertex on the circle itself, with both sides as chords. For the same arc, the inscribed angle is always exactly half the central angle — this is the Inscribed Angle Theorem.

Not meaningfully for a single arc — 360° (2π radians) already represents a full circle. An angle larger than that would just wrap around and overlap itself, so central angles are conventionally kept between 0° and 360°.

Arc length measures the curved path along the circle's edge, while the chord is the straight segment connecting the same two points. They're equal only in the limiting case of a vanishingly small angle; for any actual angle, the arc is always slightly longer than the chord.

The arc length becomes the full circumference of the circle, since s = rθ = r × 2π = 2πr — exactly the standard circumference formula, confirming the two relationships agree at the boundary case.

Yes — the magnitude of the arc length only depends on the size of the angle, not its rotational direction, so s = rθ applies the same way regardless of which direction the angle opens.