Perimeter of a Sector Calculator

Enter a sector's radius and central angle to get its perimeter (P = 2r + rθ), arc length (L = rθ), and area (A = ½r²θ).

Quick Facts

Perimeter formula
P = 2r + rθ (θ in radians)
Two straight radii plus the curved arc length.
Arc length formula
L = rθ
Just the curved outer edge of the sector.
Sector area formula
A = ½r²θ
θ must be in radians for this formula to work directly.
Degrees to radians
θ(rad) = θ(deg) × π/180
Example: 90° = π/2 ≈ 1.5708 rad.

Your Results

Calculated
Perimeter
-
P = 2r + rθ
Arc Length
-
L = rθ
Sector Area
-
A = ½r²θ
Angle in Radians
-
θ converted to radians

Ready

Enter a radius and central angle, then press Calculate.

Formula and Method for the Perimeter of a Sector

A circular sector is the pie-slice-shaped region bounded by two radii and the arc between them. Its perimeter is the total distance around that boundary: the two straight radii plus the curved arc length. The formula is P = 2r + rθ, where r is the radius and θ is the central angle expressed in radians. This calculator also derives the arc length and sector area from the same inputs, and accepts the angle in either degrees or radians.

How the calculation works

Enter the radius and the central angle, and choose whether the angle is in degrees or radians. If you enter degrees, the calculator first converts to radians using θ(rad) = θ(deg) × π/180. It then multiplies the radius by the angle in radians to get the arc length (L = rθ), adds the two straight radii (2r) to get the perimeter (P = 2r + rθ), and multiplies half the radius squared by the angle in radians to get the sector area (A = ½r²θ). All three results use the radian form directly, since the rθ and r²θ relationships only hold when θ is in radians.

Common mistakes

  • Forgetting to convert degrees to radians: plugging a degree value straight into rθ or ½r²θ gives a badly wrong answer — always convert first, or use the degree form P = 2r + (θ/180) × πr.
  • Confusing perimeter with arc length: the arc length (rθ) is only the curved edge. The full perimeter must also include the two straight radii: P = 2r + rθ.
  • Angle greater than 360° (or 2π radians): a sector's central angle cannot exceed a full circle — check your input if the calculator flags it as out of range.

Real-world applications

  • Cutting a pie or pizza slice — the crust edging length is the arc length, while the total edge (crust plus two straight cuts) is the perimeter.
  • Sprinkler and irrigation design uses sector perimeter and area to plan coverage for rotating sprinkler heads.
  • Fan-shaped garden beds, patios, and window designs use the perimeter to estimate edging or trim material.
  • Mechanical and civil engineering use sector geometry for gear teeth, road curves, and pipe fittings.

Frequently Asked Questions

What is the formula for the perimeter of a sector?
The perimeter of a circular sector is P = 2r + rθ, where r is the radius and θ is the central angle in radians. The two straight radii contribute 2r, and the curved edge (the arc length) contributes rθ. In degrees, this is P = 2r + (θ/180) × πr.
Do I need to convert the angle to radians first?
Only if you are applying the rθ form of the formula by hand. Multiply degrees by π/180 to get radians (for example, 90° = π/2 ≈ 1.5708 rad). This calculator does the conversion for you when you choose degrees.
What is the difference between arc length and perimeter?
Arc length (L = rθ) is only the curved outer edge of the sector. Perimeter adds the two straight radii to the arc length: P = 2r + L. The full sector boundary is a closed shape, so both radii must be included.
What is the perimeter of a semicircle?
A semicircle is a sector with a 180° (π radian) central angle, so its perimeter is P = 2r + πr = r(2 + π). For a radius of 5, that is 5 × (2 + π) ≈ 25.71.

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