Arc Length & Sector Area Calculator

Calculate arc length and sector area of a circle from radius and central angle. Arc length = rθ (in radians). Sector area = ½r²θ. Includes degree-to-radian conversion.

📐 Trigonometry📐 L = rθ (θ in radians)🔢 Math
Radius r
Central angle
Angle unit
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Arc Length & Sector Area Calculator—L = rθ (θ in radians)same as radius units

Step-by-Step Examples

Example 1
Arc at 60°

r=5, 60°.

  • θ=60×π/180=π/3
  • L=5×π/3≈5.236
  • Area=½×25×π/3≈13.09
✓ L ≈ 5.24, Area ≈ 13.09
Example 2
Semicircle

r=4, 180°.

  • L=4×π≈12.57 (half circumference)
  • Area=½×16×π≈25.13 (half circle)
✓ L = πr ≈ 12.57
Example 3
Full Circle

r=3, 360°.

  • L=3×2π≈18.85 (circumference)
  • Area=½×9×2π≈28.27 (full circle)
✓ L = 2πr ≈ 18.85

Real-World Applications

🚗
Engineering
Road curves, railroad curves, pipeline bends — arc length for material calculations.
🎮
Game Dev
Circular movement, camera rotation arcs, projectile curves.
📐
Geometry
Sectors, segments, and arcs appear in many geometric constructions.
📊
Calculus
Arc length formula in calculus: L=∫√(1+(dy/dx)²)dx generalizes this.

Common Mistakes to Avoid

⚠️
Angle must be in radians for L=rθ

The formula L=rθ requires θ in radians. If given degrees, convert first: θ_rad = θ_deg × π/180.

⚠️
Arc length ≠ chord length

Arc length follows the curve. Chord length is the straight line between endpoints. L=rθ for arc; chord = 2r×sin(θ/2).

⚠️
Sector area not triangle area

Sector area = ½r²θ (pie slice). Triangle area formed by radii = ½r²sin(θ). Sector = triangle + circular segment.

Frequently Asked Questions

What is an arc? ▾
A portion of the circumference of a circle. Arc length L = rθ where θ is the central angle in radians.
What is a sector? ▾
A pie-slice region bounded by two radii and an arc. Area = ½r²θ or equivalently (θ/2π)×πr².
What is a radian? ▾
One radian is the angle that subtends an arc equal to the radius. So for angle θ radians, arc length = r×θ — this is why radians are natural.
What is a circular segment? ▾
The region between a chord and the arc it cuts off. Area = sector area − triangle area = ½r²(θ−sin(θ)).
Arc length in calculus? ▾
For a curve y=f(x): L=∫√(1+(f'(x))²) dx from a to b. For parametric: L=∫√((dx/dt)²+(dy/dt)²) dt.

Related Math Calculators

📐 Sector Area📐 Trig Functions📐 Degrees/Radians📐 Circle📐 Ellipse

Formula Explorer connections

Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.

Area Calculator – Area of Any 2D Shape →Area of an Ellipse Calculator →Area of a Parallelogram Calculator →Math Formula Explorer →