Sector Area Calculator

Enter a circle's radius and central angle to get the sector's area, arc length, chord length, and perimeter.

Quick Facts

Sector area (radians)
A = ½r²θ
θ must be in radians; multiply degrees by π/180 first.
Sector area (degrees)
A = (θ/360) × πr²
Direct formula when the angle is given in degrees.
Arc length
L = rθ (rad) = (θ/360) × 2πr
The curved edge length — not the straight-line chord.

Your Results

Calculated
Sector Area
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A = ½r²θ (radians)
Arc Length
-
L = rθ (radians)
Chord Length
-
c = 2r × sin(θ/2)
Sector Perimeter
-
P = 2r + arc length

Ready

Enter a radius and central angle, then press Calculate.

Formula and Method for Sector Area

A circular sector is the pie-slice-shaped region bounded by two radii and the arc between them. Its area is the fraction of the full circle's area that corresponds to the central angle θ: with θ measured in radians, A = ½r²θ; with θ measured in degrees, A = (θ/360) × πr². This calculator also derives the arc length, the straight-line chord connecting the two endpoints, and the sector's total perimeter from the same radius and angle.

How the calculation works

Enter the radius, its unit, the central angle, and whether that angle is in degrees or radians. If you choose degrees, the calculator first converts to radians by multiplying by π/180 (θ_rad = θ_deg × π/180), since the core formulas are defined in radians. It then computes sector area as A = ½r²θ, arc length as L = rθ (the curved distance along the edge), chord length as c = 2r × sin(θ/2) (the straight line between the arc's endpoints, from the isosceles triangle formed by the two radii), and sector perimeter as P = 2r + L (the two straight radii plus the curved arc).

Common mistakes

  • Mixing angle units: a central angle of 60 typed as radians is treated as 60 radians (more than 9 full turns), not 60°. Always match the angle value to the selected unit.
  • Arc length vs. chord length: the arc follows the curve and is always longer than the straight-line chord (except at very small angles, where they nearly coincide). Do not use one where the other is required.
  • Angle greater than 360° (or 2π radians): a sector cannot sweep more than a full circle, so angles beyond 360° or 2π ≈ 6.2832 radians are outside the valid domain for this calculator.

Real-world applications

  • Pie charts and dashboards use sector area to size wedges proportionally to the data they represent.
  • Sprinkler and irrigation layout uses sector area and arc length to plan coverage for rotating heads that water less than a full circle.
  • Machining and sheet-metal work uses arc length and chord length to lay out curved cuts, gears, and pipe fittings.
  • Architecture and landscape design use sector geometry for curved patios, fan-shaped rooms, and road-curve layouts.

Frequently Asked Questions

What is the formula for the area of a sector?
When the central angle θ is in radians, sector area equals A = ½r²θ. When θ is in degrees, use A = (θ/360) × πr². Both give the same result — a sector is just the fraction θ/360 (or θ/2π) of the full circle's area.
How do I find the arc length of a sector?
Arc length equals L = rθ when θ is in radians, or L = (θ/360) × 2πr when θ is in degrees. This is the curved distance along the circle's edge, not the straight-line distance between the two endpoints.
What is the difference between arc length and chord length?
Arc length follows the curve of the sector, while the chord is the straight line segment connecting the two endpoints of the arc: chord = 2r × sin(θ/2). The chord is always shorter than the arc length for angles less than 360°.
How do I convert degrees to radians for this calculation?
Multiply the angle in degrees by π/180 (approximately 0.0174533) to get radians. For example, 60° × π/180 ≈ 1.0472 radians. This calculator handles the conversion automatically based on the angle unit you select.

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