The "perimeter" of a sector (sometimes called its arc length perimeter or the perimeter of a circular sector) is the total distance around the sector's boundary — this includes the curved arc plus the two straight radii that form the sides of the "pie slice."
Formula:
Perimeter = 2r + arc length
where:
- r = radius of the circle
- arc length = (θ/360°) × 2πr, if the angle θ is in degrees
- arc length = θ × r, if the angle θ is in radians
Putting it together, the full formulas are:
- In degrees: Perimeter = 2r + (θ/360°) × 2πr
- In radians: Perimeter = 2r + θr = r(2 + θ)
Step-by-step method:
- Identify the radius (r) of the circle and the central angle (θ) of the sector.
- Calculate the arc length using the appropriate formula based on whether θ is in degrees or radians.
- Add twice the radius (2r) to the arc length, since a sector has two straight edges (the two radii) plus the curved arc.
Example: Suppose a sector has a radius of 10 cm and a central angle of 60°.
- Arc length = (60/360) × 2π(10) = (1/6) × 20π ≈ 10.47 cm
- Perimeter = 2(10) + 10.47 = 20 + 10.47 = 30.47 cm
Key notes:
- Make sure the angle unit (degrees vs. radians) matches the formula you use — mixing them up is the most common mistake.
- If you're only given the arc length directly (not the angle), you can skip calculating it and just add it directly to 2r.
- This perimeter calculation applies to a "minor" or "major" sector alike — the method is the same regardless of whether the angle is less than or greater than 180°.