How this calculation works
Use one common center and a positive angle through a full turn. For partial sectors add two radial cut edges; at 360° the region closes and those cut edges are no longer boundary.
θ = degrees × π/180; area = θ(R²−r²)/2; boundary = (R+r)θ plus exposed radial cutsA worked example
Outer radius 5 m, inner radius 2 m and angle 90° give area 21π/4 ≈ 16.493361 m². Total boundary is 7π/2 + 6 m, including two exposed 3 m radial cuts.
Good to know
Concentric circular regions only, with 0 ≤ inner radius < outer radius. Zero-angle degeneracy is excluded. Boundary is for the closed mathematical region, without cut seams, overlap or manufacturing allowances.
Reference: MathWorld: geometry definitions and formulas (opens in a new tab) ↗
Removing the inner sector changes area and boundary differently
The default outer radius 5 m and inner radius 2 m leave an annular band. A 90° angle is π/2 radians, giving area (π/2)(25−4)/2 = 21π/4, about 16.493361 m². Setting the inner radius to zero instead gives a filled circular sector.
Its outer arc is 5π/2 m and inner arc π m. The two exposed radial cuts are each 3 m long, so total boundary is 7π/2 + 6 m. Subtracting inner area does not mean subtracting the inner boundary length: the hole-side arc is also part of the boundary.
The circle calculator gives a complete filled disk. A partial sector has exposed radial cuts, while an exact full turn does not.
At a full turn the cut edges stop being exposed
At 360°, the same radii give a complete annulus. Its boundary consists only of the two circles, totaling 2π(5+2) = 14π m. There are no exposed radial cut edges in the closed mathematical region.
A nearly full partial sector still has two cuts, while the exact closed turn has none. The boundary-length convention therefore changes at closure. A manufacturing seam or intentional cut may still exist in a physical part, but it is not automatically included as region perimeter.
The angle is entered in degrees and converted once
Arc formulas use radians. The calculator explicitly converts degrees × π/180 before multiplying by radius. Entering 90 as radians in a degree-based calculation would produce a different and invalid interpretation of the intended quarter turn.
The circle calculator covers a full filled disk. The ring torus calculator describes a three-dimensional circular tube swept around an axis, rather than this planar annular region. All radius numbers on this page must use the same common length unit.
A few questions, answered
Is the inner arc subtracted from perimeter?
No. It is a boundary of the annular region and is included along with the outer arc.
Why are no radial cuts counted at 360°?
A full disk or annulus is closed without exposed radial cut edges. A physical seam is a different allowance.
Can I enter a zero angle?
No. This model excludes the zero-angle degenerate region and accepts positive angles through 360°.
Can the inner radius equal the outer radius?
No. A positive-width sector or annular sector requires inner radius strictly below outer radius.