Central Angle Calculator
This calculator works out the central angle of a circle - the angle formed at the centre by two radii - using whichever measurements you already have. Choose between two input methods: enter an arc length and radius, or enter a sector area and radius, and the tool solves for the angle behind them. You can also choose whether the headline result displays in degrees or radians, and set how many decimal places to show. Once you enter your figures, you get the central angle as the highlighted result in your chosen unit, alongside the same angle expressed in radians, the arc length and the sector area. Below that, a fuller breakdown lists every related sector measurement: the radius, the angle in both degrees and radians, arc length, sector area, chord length and sector perimeter, plus a formula verification panel showing exactly which method was used, the working behind the answer, the full circle's circumference, what fraction of the circle your arc represents, and a check calculation confirming arc length equals radius times angle in radians. This makes the tool useful for checking pie-slice areas, road curve angles, sprinkler sweep angles, or any situation where you know two of arc length, radius, angle and area and need the rest. Keep arc length and radius in the same unit; this uses standard Euclidean geometry, so results are exact, not estimates.
Input Method
Angle Unit Display
Full Sector Measurements
Formula Verification
Worked Example (default inputs)
Given arc length = 7.854 units and radius = 5 units:
Central angle (θ) = arc length / radius = 7.854 / 5 = 1.5708 radians
Convert to degrees: 1.5708 × (180 / π) = 90.000°
Sector area = ½ × r² × θ = 0.5 × 25 × 1.5708 = 19.635 square units
Chord length = 2r × sin(θ/2) = 10 × sin(45°) = 7.071 units
Sector perimeter = arc + 2r = 7.854 + 10 = 17.854 units
What Is a Central Angle?
A central angle is an angle formed at the centre of a circle, with its two sides being radii that extend to the circumference. The arc of the circle that lies between those two radii is called the intercepted arc. The central angle and its intercepted arc have the same angular measure: a central angle of 90 degrees subtends exactly one quarter of the circle's circumference.
Central angles are fundamental to circle geometry and appear in many practical applications: from calculating the area of a pie slice (sector), to describing the sweep of a sprinkler, to finding the angle subtended by a road curve.
Central Angle Formulas
There are two standard ways to calculate the central angle, depending on what information you have.
| Known values | Formula (radians) | Formula (degrees) |
|---|---|---|
| Arc length (s) and radius (r) | θ = s / r | θ = (s / r) × (180 / π) |
| Sector area (A) and radius (r) | θ = 2A / r² | θ = (2A / r²) × (180 / π) |
| Fraction of circle (f) | θ = 2πf | θ = 360 × f |
The arc length formula (θ = s / r) comes directly from the definition of a radian: one radian is the angle at the centre of a circle where the arc length equals the radius. This makes radian measure the natural unit for circular geometry.
Related Circle Measurements
Once you know the central angle and radius, you can find every other measurement of the circular sector:
- Arc length: s = r × θ (where θ is in radians)
- Sector area: A = ½ × r² × θ (where θ is in radians)
- Chord length: c = 2r × sin(θ / 2)
- Sector perimeter: P = s + 2r (arc plus the two radii)
Degrees vs Radians
Degrees divide a full circle into 360 equal parts. Radians relate the angle to the arc length and radius directly: a full circle is 2π radians (approximately 6.2832 radians). The conversion is simple: multiply radians by 180/π to get degrees, or multiply degrees by π/180 to get radians.
| Degrees | Radians | Description |
|---|---|---|
| 30° | π/6 ≈ 0.5236 | One twelfth of a circle |
| 45° | π/4 ≈ 0.7854 | One eighth of a circle |
| 60° | π/3 ≈ 1.0472 | One sixth of a circle |
| 90° | π/2 ≈ 1.5708 | Quarter circle (right angle) |
| 180° | π ≈ 3.1416 | Half circle (straight angle) |
| 270° | 3π/2 ≈ 4.7124 | Three-quarter circle |
| 360° | 2π ≈ 6.2832 | Full circle |
Related Calculators
- Maths and Stats Calculators: the full hub for geometry and maths tools.
- Arc Length and Sector Area Calculator: find the arc and sector from angle and radius.
- Circumference Calculator: calculate the perimeter of a full circle.
- Area of a Circle Calculator: find the area of a full circle from its radius or diameter.
- Annulus Calculator: area and perimeter of a ring between two concentric circles.
Method: Standard circle geometry. Central angle from arc length: θ (rad) = s / r; convert to degrees by multiplying by 180/π. Central angle from sector area: θ (rad) = 2A / r². Chord length: c = 2r sin(θ/2). Sector perimeter: P = s + 2r. All formulas are standard Euclidean geometry.