Annular Sector Area Calculator
From an inner radius, an outer radius, and a central angle, get the area of the slice and the lengths of its two arcs.
Three inputs, three answers
Enter the inner radius r₁, the outer radius r₂, and the central angle and the calculator returns the area, the outer arc length, and the inner arc length at once.
Outer radius beats inner
A ring needs the outer radius larger than the inner radius, and the angle stays between 0° and 360° — a full ring is exactly 360°.
What is an annular sector area calculator?
Two radii and an angle in, full slice out
An annular sector area calculator turns three measurements — the inner radius r₁, the outer radius r₂, and the central angle — into the numbers that describe a slice of a ring: the area of the slice, the length of its outer arc, and the length of its inner arc. An annular sector is what you get when you cut a wedge from a washer or a CD: a curved band bounded by two radii and two arcs. That makes the calculator handy for washers, curved ramps, fan blades, pie-chart rings, arched windows, and any geometry problem with a ring slice in it.
Enter the inner radius, outer radius, and central angle to get the area and both arc lengths instantly.
Three short formulas, all built from the two radii and the angle. The angle is first converted from degrees to radians with θ = angle × π/180.
area = (θ/2) × (r₂² − r₁²)The area is half the angle (in radians) times the difference of the squared radii: (θ/2) × (r₂² − r₁²). The outer arc length is the outer radius times the angle in radians, r₂ × θ, and the inner arc length is r₁ × θ. Because the angle has to be in radians, the degrees you type are converted first — 90° becomes π/2 ≈ 1.5708.
Suppose you have a ring slice with an inner radius of 3, an outer radius of 5, and a 90° angle.
Outer arc length
θ = 90 × π/180 = π/2, so r₂ × θ = 5 × π/2 = 7.853982 — the long curved edge.
Inner arc length
r₁ × θ = 3 × π/2 = 4.712389 — the short curved edge.
Area
(θ/2) × (5² − 3²) = (π/4) × 16 = 12.566371 square units — the area of the slice.
The three outputs answer three different everyday questions. The area (about 12.566371 square units for r₁ = 3, r₂ = 5, 90°) is the size of the band itself — the metal in a washer wedge, the surface of a curved ramp, the slice of a doughnut-shaped chart. Notice it depends on the difference of the squared radii, not the radii directly, so a slice near the outer rim covers far more area than the same angle near the centre. The outer arc length (7.853982 here) is the long curved edge you would trim or trace along the outside; the inner arc length (4.712389) is the shorter curved edge on the inside. Both arcs grow in direct proportion to the angle, and the outer arc is always longer because it sits at the bigger radius. The single most useful insight is that the angle drives everything linearly through θ — double the angle and both arcs and the area double too — while the radii set the scale. Keep the outer radius above the inner radius and the angle inside 0–360° and the slice is well defined.
The formulas are exact, but a couple of practical points are worth keeping in mind.
Valid rings, angle range, and consistent units
These formulas describe a flat annular sector — a slice of a true ring — so the outer radius must be strictly larger than the inner radius, otherwise there is no band and the calculator returns nothing. The central angle must lie between 0° and 360°; a value above 360° would wrap past a full ring and is rejected. The radii are unit-agnostic, so keep one unit throughout: radii in centimetres give an area in square centimetres and arc lengths in centimetres. The angle, however, is always in degrees — it is not a length, so it never shares the radii's unit.