Geometry
Central Angle Calculator
Central angle from arc length and radius (degrees).
Free to use — no sign-up or login.
θ = central angle · s = arc length · r = radius
Angle (rad)
Angle (°)
Amortization schedule
| # | Date | Payment | Principal | Extra | Interest | Balance |
|---|---|---|---|---|---|---|
This free central angle calculator reports the opening at the center of a circle from intercepted arc length s and radius r. The defaults — s = 4 and r = 5 — give θ = 4/5 = 0.8 radians, which is 0.8 × (180/π) ≈ 45.84°. A central angle has its vertex at the center O. Its two sides are radii from O to the endpoints of the intercepted arc. The arc length s is the rim distance between those endpoints, not the straight chord.
No account is required. How it works beside the form keeps θ = s/r (radians) and θ° = (s/r) × (180/π) on separate lines. You type s and r; the page reports Angle (rad) and Angle (°). It does not solve for s or r. Arc length from a known angle is the Arc Length Calculator.
What is a central angle?
A central angle is an angle whose vertex is the center of a circle. Each side is a radius: a segment from O to a point on the circumference. Those two radii meet the circle at the endpoints of the intercepted arc — the portion of the rim that lies between them. Radius r is the common length of those sides. Arc length s is measured along the rim, in the same length unit as r. An inscribed angle that intercepts the same arc has its vertex on the circumference instead and measures half as much; this widget is only the central case, with the vertex fixed at O.
Central angle formula — radians and degrees
In radians, the central angle is the ratio of arc length to radius: θ = s/r. That identity is the definition of a radian: θ = 1 rad when s = r. Degrees are another unit for the same opening: θ° = (s/r) × (180/π). With the defaults s = 4 and r = 5, θ = 0.8 rad and θ° ≈ 45.84°. Rearranged, s = rθ only when θ is in radians. This page does not take a typed degree input; it always starts from s and r, then prints both units.
How to use this calculator
Type Arc length s (default 4) and Radius r (default 5). Angle (rad) is s/r; Angle (°) is that quantity times 180/π. The two length inputs must use the same unit — centimeters of arc with centimeters of radius. You do not enter θ. Disk area from a known r is the Area of a Circle Calculator and the How to Calculate Circle Area and Circumference tutorial, not this angle widget.
Semicircle, full circle, and θ = 0
If s = 0 and r is nonzero, θ = 0: the two radii coincide and there is no intercepted arc. A semicircle intercepts half the circumference, so s = πr and θ = π rad = 180°. A full turn is the whole circumference s = 2πr, so θ = 2π rad = 360°. At the default r = 5 those arcs are s = 5π ≈ 15.71 (180°) and s = 10π ≈ 31.42 (360°). The Semicircle Area Calculator is the half-disk area, not this central-angle identity.
Invalid radius and arcs longer than the circumference
Radius r = 0 cannot divide: the script throws "r cannot be zero." A negative r is not blocked in the formula — θ = s/r can come out negative when s and r have opposite signs. The live figure draws only for r > 0. If s is longer than the circumference 2πr, θ exceeds 2π (360°). The code does not wrap into [0, 360): s = 4πr would report 4π rad = 720°. A negative s yields a negative angle, the other way around the circle.
Related circle formulas
Area πr² and circumference 2πr from a known radius are the Area of a Circle Calculator. Arc length from radius and a typed angle is the Arc Length Calculator. Expanding (x − h)² + (y − k)² = r² is the Equation of a Circle Calculator. A half-disk is the Semicircle Area Calculator. A tangent segment from an external point to a circle about the origin is the Tangent of a Circle Calculator. Sine and cosine as a point on x² + y² = 1 are the Unit Circle Calculator.
Worked example: arc length 4 and radius 5
Arc length s = 4 and radius r = 5. The central angle in radians is θ = s/r = 4/5 = 0.8. In degrees, θ° = 0.8 × (180/π) ≈ 45.84°. Those are the Angle (rad) and Angle (°) fields for the defaults. The intercepted arc is the rim of length 4 between the two radii; it is shorter than the circumference 2πr ≈ 31.42, so the angle is well under a full turn.
\[ \theta=4/5=0.8,\quad \theta^\circ=0.8\times(180/\pi)\approx 45.84^\circ \]
Frequently asked questions
How do I find the central angle of a circle?
Divide the intercepted arc length by the radius: θ = s/r in radians. Convert with θ° = (s/r) × (180/π). Defaults s = 4 and r = 5 give 0.8 rad ≈ 45.84°.
What is the difference between radians and degrees here?
Both numbers describe the same central angle. Radians are s/r; degrees are that ratio times 180/π. One full turn is 2π rad or 360°. This page prints both from the same s and r.
What is an intercepted arc?
The intercepted arc is the portion of the circumference lying between the two radii that form the central angle. Its length is s; the radii themselves have length r and meet at the center O.
What happens at 0°, 180°, and 360°?
s = 0 gives θ = 0 (the radii coincide). s = πr gives 180°, a semicircle. s = 2πr gives 360°, a full circle. At r = 5 those arcs are 0, 5π ≈ 15.71, and 10π ≈ 31.42.
What if radius r is zero or negative?
r = 0 throws "r cannot be zero." A negative radius is not rejected: θ = s/r can be negative when s and r have opposite signs. The figure draws only when r is positive.
Can the arc be longer than the circumference?
Yes. If s > 2πr the reported angle is greater than 360°. The formula does not reduce modulo 2π, so s = 4πr yields 720° rather than wrapping to 0°.
References
Assumptions and limitations
These central-angle results are unofficial. π is the same constant as JavaScript Math.PI. Length units on s and r must match; the angle is a pure number in radians or degrees.