About

A sector is the region enclosed by two radii and the connecting arc of a circle. Miscalculating sector area propagates errors into material estimates for curved panels, irrigation coverage, radar sweep zones, and pie-chart proportions. This calculator applies A = 12 r² θ (radians) or the degree-equivalent form, deriving arc length, perimeter, and chord simultaneously. Results assume a perfect Euclidean plane. For sectors on curved surfaces the formula breaks down; apply spherical geometry corrections in those cases.

Precision matters: a 1% error in r squares into a ≈ 2% error in area because area scales with r². Always measure radius at the midpoint of material thickness and verify your angle instrument reads true degrees, not grads. Pro tip: if you only know the arc length and radius, divide L by r to recover θ in radians before computing area.

Formulas

The area of a sector with central angle in degrees:

A = θ360 × π × r²

When the central angle is given in radians:

A = 12 × r² × θ

Arc length of the sector:

L = θ360 × 2 × π × r

Perimeter (total boundary length) of the sector:

P = 2r + L

Chord length connecting the two radii endpoints:

c = 2r × sin(θ2)

Where A = sector area, r = radius, θ = central angle, L = arc length, P = sector perimeter, c = chord length, π ≈ 3.14159265.

Reference Data

Central Angle (θ)	Fraction of Circle	Area Factor (× πr²)	Arc Length Factor (× 2πr)	Common Name
15°	124	0.04167	0.04167	Hour on clock face
30°	112	0.08333	0.08333	Clock hour sector
45°	18	0.12500	0.12500	Octant
60°	16	0.16667	0.16667	Sextant
90°	14	0.25000	0.25000	Quadrant / Right angle
120°	13	0.33333	0.33333	Third of circle
135°	38	0.37500	0.37500	Three octants
150°	512	0.41667	0.41667	Five twelfths
180°	12	0.50000	0.50000	Semicircle
210°	712	0.58333	0.58333	Major sector (reflex start)
240°	23	0.66667	0.66667	Two thirds
270°	34	0.75000	0.75000	Three quadrants
300°	56	0.83333	0.83333	Five sixths
330°	1112	0.91667	0.91667	Eleven twelfths
360°	1	1.00000	1.00000	Full circle

Frequently Asked Questions

A sector is the "pie slice" bounded by two radii and the arc between them. A segment is the region between a chord and the arc it subtends. To get the segment area, compute the sector area using A = 12r²θ then subtract the triangle area 12r²sinθ.

Area scales with the square of the radius. Doubling r quadruples A. For example, a sector with r = 5 and θ = 90° has area 19.635. At r = 10 the area is 78.540, exactly 4× larger.

Geometrically a sector angle is in the range 0° < θ ≤ 360°. An angle of 360° yields the full circle. This calculator clamps the angle to that range. In engineering contexts, angles above 360° typically represent cumulative rotation, not a single sector.

Use radians when working with calculus-based problems, physics (angular velocity ω is in rad/s), or any formula that derives from arc length s = rθ. Degrees are conventional in construction, surveying, and everyday measurement. The calculator converts internally so both give identical results.

Because A ∝ r², relative error approximately doubles. If your radius has ±2% uncertainty, expect roughly ±4% uncertainty in area. For high-precision work (machined parts, optical components), measure radius to at least one extra significant figure beyond your area tolerance.

No. This tool assumes a perfect circle with uniform radius. An elliptical sector requires integration of 12r(θ)² dθ where r(θ) follows the ellipse polar equation. For slight eccentricity, the circular approximation using the average of semi-axes gives a rough estimate within 5%.