Central Angle, Arc Length & Radius — Circle Geometry
Calculate central angle, arc length, or radius from the other two. The formula s = rθ (θ in radians) connects arc length, radius, and angle. Supports degrees and radians.
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Central angle has vertex at the center of the circle s = rθ requires θ in radians — convert degrees: θ_rad = θ_deg × π/180 Sector area A = ½r²θ (θ in radians) Earth time zones: 24 zones × 15° = 360° Great circle distance = Earth radius × central angle (radians)
Ready to run the numbers?
Why: Central angles are essential in navigation (great circle distance), time zones (15° per hour), pie charts, gears, and any circular or rotating system. The radian makes s = rθ elegant.
How: The calculator uses s = rθ (θ in radians). Convert degrees to radians with θ_rad = θ_deg × π/180. Find angle from arc and radius, arc from angle and radius, or radius from arc and angle.
Run the calculator when you are ready.
Central Angle Calculator
Calculate central angle, arc length, or radius. Enter two values and find the third. Supports degrees and radians.
📐 Sample Examples — Click to Load
Calculation Mode
Input
Properties Comparison
Arc & Sector Proportions
Properties Radar (Scaled)
Step-by-Step Breakdown
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
Earth is divided into 24 time zones by 15° central angles — each zone spans 15° of longitude
— Geography
The radian is defined so that s = rθ: when θ = 1 rad, arc length equals radius
— Wolfram MathWorld
Great circle distance between cities uses the central angle from Earth's center — NYC to Tokyo ≈ 98.5°
— Navigation
Pie charts use central angles to represent proportions — 25% of data = 90° sector
— Data visualization
Key Takeaways
- Central angle is formed at the center of a circle by two radii — vertex at the center
- Arc length formula: s = r × θ (θ must be in radians)
- Central angle from arc: θ = s / r (result in radians; multiply by 180/π for degrees)
- Radius from arc and angle: r = s / θ
- Sector area: A = ½r²θ (θ in radians) or A = πr² × (θ°/360°)
Did You Know?
How Central Angles Work
The central angle θ (in radians) relates arc length s and radius r. Always use radians for s = rθ.
Arc Length: s = r × θ
Multiply radius by the central angle in radians. Convert degrees: θ_rad = θ_deg × π/180.
Central Angle: θ = s / r
Divide arc length by radius. Result is in radians. Convert to degrees: θ_deg = θ_rad × 180/π.
Radius: r = s / θ
Divide arc length by the central angle (in radians).
Sector Area: A = ½r²θ
The sector is the region bounded by the central angle and its arc. Proportional to θ.
Expert Tips
Always Use Radians for s = rθ
The formula s = rθ requires θ in radians. Convert: θ_rad = θ_deg × π/180.
Arc Must Be Less Than Circumference
For angle mode: s < 2πr. Otherwise the "angle" would exceed 2π (full circle).
Special Angles
90° = π/2 rad (quarter circle), 180° = π (semicircle), 360° = 2π (full circle).
Great Circle Navigation
Distance = R × θ (radians). Earth R ≈ 6371 km. NYC–Tokyo: θ ≈ 98.5° → ~10,940 km.
Formula Comparison
| Find | Formula | Given |
|---|---|---|
| Arc length | s = r × θ | r, θ (rad) |
| Central angle | θ = s / r | s, r |
| Radius | r = s / θ | s, θ (rad) |
| Sector area | A = ½r²θ | r, θ (rad) |
Frequently Asked Questions
Why must the angle be in radians for s = rθ?
The radian is defined so that arc length = radius when θ = 1 rad. This makes the formula dimensionally consistent and simple.
How do I convert degrees to radians?
Multiply by π/180. So 90° = 90 × π/180 = π/2 ≈ 1.5708 rad.
What is the sector area formula?
A = ½r²θ (θ in radians). Equivalently, A = πr² × (θ°/360°) — the sector is (θ/2π) of the full circle.
Can the arc length exceed the circumference?
For a single arc, typically θ ≤ 2π. If θ > 2π, you have more than one full rotation.
What is the difference between central and inscribed angles?
Central angle has vertex at the center; inscribed angle has vertex on the circumference. Inscribed angle = ½ × intercepted arc.
How is this used in navigation?
Great circle distance = Earth radius × central angle (radians). The central angle between two points on Earth gives the shortest path.
What are applications in engineering?
Gears, pulleys, conveyor belts, curved structures, lens design, and any rotating or circular system use central angle relationships.
How accurate is this calculator?
Uses JavaScript Math.PI. Results shown with 2 decimal places. For critical applications, verify with domain tools.
Central Angle by the Numbers
Official & Trusted Sources
Disclaimer: This calculator uses standard formulas s = rθ, θ = s/r, r = s/θ, and A = ½r²θ. Results are limited by floating-point precision. For critical engineering or navigation applications, verify with domain-specific tools. Not a substitute for professional analysis.
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