How this instrument works
A sector's perimeter is not simply an arc length. Cut a wedge out of a circle and its boundary has three pieces: two straight cuts running from the center out to the rim, plus the curved edge between them. P = rθ + 2r keeps those pieces separate — rθ for the curve, 2r for the two straight radii — because a full circle's rim closes on itself with no cut, while a sector's rim does not.
The rθ term follows directly from how a radian is defined: one radian is the angle at which an arc has exactly the same length as the radius. Scale that angle by θ and the arc scales by the same factor, giving arc length rθ with no extra constant to carry — a definition built for exactly this multiplication. Degrees need a conversion first for the same reason a ruler marked in inches needs converting before it works inside a metric formula.
Push θ to zero and the two straight edges fold onto each other, leaving a bare perimeter of 2r with no curve at all — the sector has collapsed to a doubled line segment. Push θ to a full turn and the wedge becomes indistinguishable from the whole disk, yet the formula still adds 2r for a seam that no longer physically exists, which is why P = rθ + 2r at a full turn comes out a touch larger than the plain circumference 2πr. Sector area, ½r²θ, answers a different question entirely: how much surface the wedge encloses, not how far its edge runs.
- Enter the wedge's Radius — the straight-line distance from the center to the rim, in any length unit you like.
- Enter Central angle and pick its unit from the selector next to the field: degrees, radians, or turns all work.
- Read Perimeter for the total boundary length: the curved arc plus the two straight radius edges, in the same unit as Radius.
- To sanity-check a full circle, set Central angle to 360 degrees and compare Perimeter against the ordinary circumference plus 2r.
Worked example — radius 5, angle 1.2 radians
Picture a pie-slice wedge cut from a circle of radius 5, its central angle set to 1.2 radians — close to 68.75°. The curved edge is arc length rθ = 5 × 1.2 = 6, measured along the rim alone, before either straight side is added in.
The two straight radius edges that close the wedge each measure r = 5, so together they add 2r = 2 × 5 = 10 to the boundary. The total perimeter is 6 + 10 = 16 exactly, with no rounding anywhere — both terms land on whole numbers for this particular angle, which is what makes 1.2 radians a clean pair to check the arithmetic against.
Questions
What is the formula for the perimeter of a sector?
P = rθ + 2r, where r is the radius and θ is the central angle in radians. The rθ term is the arc length, a direct consequence of how a radian is defined — the angle at which arc length equals radius — and 2r accounts for the two straight edges that close the wedge back at the center.
Why does the formula need θ in radians when the field on this page takes degrees?
The identity rθ only gives a true arc length when θ is radians, because a radian is defined so that arc length equals radius times angle with no extra scaling constant involved. Enter degrees or turns and the calculator converts to radians internally before it multiplies, so the field itself can stay in whichever unit is convenient for you.
How is sector perimeter different from sector area?
Perimeter measures the boundary length walked around the wedge — arc plus two straight sides, in a unit like metres. Area measures the surface enclosed inside that boundary, in squared units, using the separate formula A = ½r²θ. Both use the same r and θ but answer different questions: one is a fence, the other is the lawn it encloses.
What happens to the perimeter at a full 360° central angle?
At a full turn the wedge becomes the whole disk, and P = rθ + 2r reduces to 2πr + 2r. For radius 5 that comes to roughly 41.42 — the ordinary circumference near 31.42 plus 2r = 10 for the two radius edges, which now sit exactly on top of each other along a single cut. A true circle's boundary is just 2πr; the extra 2r here is an artifact of treating a full turn as a sector with a seam.
What is the smallest possible perimeter for a given radius?
At a zero central angle the arc vanishes and the two radius edges collapse onto each other, leaving a perimeter of exactly 2r — for radius 5, that floor is 10. Any positive central angle adds arc length on top of that minimum, so 2r is the lower bound the perimeter approaches as the wedge narrows toward a sliver.
Does the formula still work for angles greater than 360°?
Mathematically yes: θ can exceed a full turn and the arc length rθ keeps growing in step, tracing the circle's rim more than once. Geometrically a simple wedge shape stops making sense past 360°, so treat any angle beyond one full turn as describing a spiral arc length rather than a literal sector boundary.