Calculate central angle from arc length and radius, diameter, or circumference, then find arc length or inscribed angle relationships.
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Central Angle Formula
The calculator uses three formulas, one per mode.
Central angle from arc length and radius:
θ = s / r
Arc length from central angle and radius:
s = r * θ
Central Angle Theorem (inscribed angle relationship):
θ_central = 2 * θ_inscribed
- θ = central angle, in radians for the formulas above
- s = arc length, in the same units as the radius
- r = radius of the circle
- θ_inscribed = inscribed angle that intercepts the same arc
The Central angle mode divides arc length by radius to return θ. If you enter diameter, it uses r = d/2. If you enter circumference, it uses r = C / (2π). The result is shown in degrees, radians, turns, and percent of a circle.
The Arc length mode converts your angle to radians, then multiplies by the radius. Degrees become radians via θ × π/180. Turns become radians via θ × 2π.
The Inscribed angle mode applies the Central Angle Theorem. Give it an inscribed angle and it doubles. Give it a central angle and it halves.
Reference Tables
Common central angles and their equivalents.
| Degrees | Radians | Turns | Fraction of circle |
|---|---|---|---|
| 30° | π/6 | 1/12 | 8.33% |
| 45° | π/4 | 1/8 | 12.5% |
| 60° | π/3 | 1/6 | 16.67% |
| 90° | π/2 | 1/4 | 25% |
| 120° | 2π/3 | 1/3 | 33.33% |
| 180° | π | 1/2 | 50% |
| 270° | 3π/2 | 3/4 | 75% |
| 360° | 2π | 1 | 100% |
Inscribed angle to central angle pairs for the same intercepted arc.
| Inscribed angle | Central angle | Note |
|---|---|---|
| 15° | 30° | — |
| 30° | 60° | — |
| 45° | 90° | Quarter arc |
| 60° | 120° | — |
| 90° | 180° | Thales' theorem, arc is a semicircle |
Worked Examples
Example 1: Find the central angle. A pizza slice has an arc length of 12 cm along a pizza with a radius of 15 cm. Use θ = s / r.
θ = 12 / 15 = 0.8 rad. Convert to degrees: 0.8 × 180 / π ≈ 45.84°.
Example 2: Find the arc length. A central angle of 72° sits in a circle with diameter 20 m. The radius is 10 m. Convert the angle: 72 × π / 180 ≈ 1.2566 rad. Then s = r × θ = 10 × 1.2566 ≈ 12.57 m.
FAQ
What is a central angle? An angle with its vertex at the center of a circle and sides that are radii. It equals the measure of the arc it intercepts.
Do I need radians or degrees? The formula s = rθ requires radians. The calculator handles the conversion if you select degrees or turns.
How is a central angle different from an inscribed angle? A central angle has its vertex at the center. An inscribed angle has its vertex on the circle itself. For the same intercepted arc, the central angle is twice the inscribed angle.
Can the central angle be more than 360°? In standard geometry it stays at or below 360°. If your inputs produce a larger value, the arc length you entered wraps past one full revolution, and the calculator flags it.
