What is a sector of a circle?
A sector is the region of a circle bounded by two radii and the arc connecting their endpoints. The angle where the radii meet is the central angle, usually written as θ (theta).
Think of cutting a round pizza from its center. The two straight edges are radii, the curved crust is the arc, and the opening at the center is the central angle. The sector is the entire enclosed slice—not just its curved edge. This distinction matters because area, arc length and chord length measure different things.
Sector vs arc vs segment
Sector
The filled region bounded by two radii and an arc.
Arc
The curved part of the circumference between the two radii.
Segment
The region between a chord and its corresponding arc.
Why the sector-area formula works
A complete circle has area πr² and a full turn measures 360° or 2π radians. A sector simply takes the same fraction of the circle's area as its central angle takes of the full turn. Inch Calculator and GigaCalculator both express the degree version as θ/360 multiplied by πr²; in radians the expression becomes ½r²θ.
- A
- sector area
- θ
- central angle in degrees
- r
- circle radius
- A
- sector area
- θ
- central angle in radians
- r
- circle radius
The degree/radian trap: the formula ½r²θ expects θ in radians. When θ is given in degrees, use θ/360 × πr² or convert θ to radians first.
Arc length, chord length and sector perimeter
The same radius-and-angle pair determines several different geometric quantities. Keeping their definitions separate prevents one of the most common mistakes: reporting an arc length when the question asks for area.
Arc length
Use θ in radians. In degree form, the equivalent expression is s = (θ/360°)·2πr.
Chord length
The chord is the straight-line distance between the two points where the radii meet the circle. Here θ is in radians when used directly in the trigonometric formula.
Sector perimeter
A sector's perimeter consists of two radii plus the included arc. The chord is not part of the usual sector perimeter; adding the chord creates a different boundary.
How angle and radius change the area
There are two especially useful scaling laws. Holding radius constant makes sector area directly proportional to the central angle. Holding angle constant makes area proportional to the square of the radius.
Minor, semicircle, major sector and full circle
The central angle classifies the sector. The usual terminology is based on whether the sector is less than, equal to, or greater than a semicircle.
Minor sector
A central angle less than 180° forms a minor sector. Most “pizza slice” classroom examples use a minor sector.
Major sector
A central angle greater than 180° but less than 360° forms a major sector. The calculator handles these directly; the same formulas still apply.
Sector area examples
The examples below use the same relationships as the calculator, but show the arithmetic explicitly so the geometry remains visible.
r = 10 cm, θ = 60°
A 60° sector is one-sixth of a full circle.
Arc length ≈ 10.47 cm; chord ≈ 10.00 cm.
r = 8 m, θ = 90°
A quarter-circle sector is exactly one-fourth of the full circle.
The arc is 4π ≈ 12.57 m.
r = 12, θ = 1.5 rad
Here the radian formula is the shortest route.
Arc length is s = 12 × 1.5 = 18 units.
r = 5 in, θ = 270°
This sector covers three-fourths of the circle.
The major-sector arc is three-fourths of the circumference.
Reference table for common sectors
| Angle | Circle fraction | Area multiplier | Radian equivalent |
|---|---|---|---|
| 30° | 1/12 | πr²/12 | π/6 |
| 45° | 1/8 | πr²/8 | π/4 |
| 60° | 1/6 | πr²/6 | π/3 |
| 90° | 1/4 | πr²/4 | π/2 |
| 120° | 1/3 | πr²/3 | 2π/3 |
| 180° | 1/2 | πr²/2 | π |
| 270° | 3/4 | 3πr²/4 | 3π/2 |
| 360° | 1 | πr² | 2π |
Where sector geometry is used
Sectors are not only textbook shapes. They are useful whenever a circular system covers, occupies or sweeps through only part of a full rotation. The relationship between radius, angle and area appears in food portions, irrigation coverage, rotating systems and spatial planning.
A simple problem-solving method
Sector-calculator mistakes to avoid
Using degrees in a radian formula
Putting 60 directly into ½r²θ treats 60 as radians, not degrees. Convert first or use the degree formula.
Using diameter as radius
If the measured diameter is d, the radius is r = d/2. Squaring d instead of d/2 makes the area four times too large.
Confusing arc and chord
The arc is curved and follows the circle; the chord is a straight segment joining the same endpoints.
Forgetting square units
Area is measured in cm², m², in² or another squared unit. Arc and chord are measured in cm, m, in or another length unit.
Sector formulas and reference pages
The calculator and explanation above were checked against the sector-area references you supplied. These pages cover the same core relationships and examples in their own terminology.
Sector calculator FAQ
A sector is the region of a circle bounded by two radii and the included arc.
With degrees: A = (θ/360°)πr². With radians: A = ½r²θ.
Use s = rθ when θ is in radians. In degrees, use s = (θ/360°)2πr.
The chord length is c = 2r sin(θ/2), with θ measured in radians in the direct trigonometric form.
A minor sector has a central angle below 180°. A major sector has an angle above 180° and below 360°. At 180° you have a semicircle.
Yes. First convert to radius using r = d/2, then use the sector formulas.
Yes. The calculator supports both degrees and radians. Remember that one full revolution is 2π radians.
The area uses squared length units—for example cm², m², in² or ft²—while arc and chord use the original length unit.