Sector Area Calculator
A circular sector is the pie-slice region between two radii and the arc that joins them. Enter the radius and the central angle to get the sector area, arc length, chord length, and segment area. Switch between degrees and radians, choose any length unit, or use reverse mode to find the radius or angle from a known area.
Formula
Worked example
Radius 5 ft, central angle 60° (= π/3 rad): sector area = ½ × 25 × (π/3) ≈ 13.0900 ft². Arc length = 5 × π/3 ≈ 5.2360 ft. Chord length = 2 × 5 × sin(30°) = 5.0000 ft. Segment area = 13.0900 - ½ × 25 × sin(60°) ≈ 13.0900 - 10.8253 ≈ 2.2647 ft².
What a circular sector is
A circular sector is the region bounded by two radii of a circle and the arc that connects their endpoints, the shape of a pizza slice or a pie wedge. Its size depends on two things only: how long the radii are and how wide the central angle is between them. Because every sector is simply a wedge cut from a full disc, its area is the corresponding fraction of the whole circle. That fraction is the central angle divided by 2pi radians (or equivalently 360 degrees), which is why the formula is so compact. This calculator handles the full family: the sector itself, the arc along its curved edge, the straight chord that closes it, and the circular segment left when the central triangle is removed.
How the formulas work
Start with the sector area formula: A = one half r squared times theta, where theta is the central angle in radians. In degree form this becomes A = (theta-degrees / 360) times pi r squared, identical in result. Arc length is even simpler: L = r theta, since the arc is the same fraction of the full circumference 2 pi r. Chord length uses basic trigonometry: c = 2 r sin(theta / 2), connecting the two points where the radii meet the circle. Segment area strips away the isoceles triangle formed by the two radii and the chord, so A-segment = A-sector minus one half r squared sin(theta). All four outputs are computed from the same two inputs and scale consistently with any length unit.
Reverse-solve: find the radius or angle from a known area
When you know the sector area but not the radius, rearrange the formula: r = sqrt(2A / theta). When you know the area and the radius but need the angle, theta = 2A / r squared. Both reverse modes are built into this calculator under the "Solve for" selector. They are useful when fitting a sector to a target surface, sizing a pie-chart wedge to a percentage, or working backwards from measured field data. The arc, chord, and segment outputs are all updated in every mode so you get the full geometric picture regardless of which pair of values you started with.
Practical uses of sector area
Sector area appears across many fields. Pie and donut charts size each wedge by its fraction of 360 degrees. Irrigation engineers calculate the ground a rotating sprinkler sweeps. Architects and designers lay out fan shapes, gauge faces, and clock faces. Machinists use it to find the material in a curved bracket or the swept area of a rotating arm. Surveyors calculate land parcels bounded by circular arcs. Choosing the right length unit, such as metres for land, feet for building, or centimetres for machining, keeps the result in the unit you need without manual conversion.
Degrees versus radians: when to use each
Degrees are the everyday unit for angles in construction, navigation, and most engineering drawings. Radians are the natural unit in mathematics and physics because they make the arc length formula simply L = r theta without any conversion factor. This calculator accepts both: select "Degrees" for angles like 90, 120, or 270, and "Radians" if your data comes from a formula or trigonometric computation. If you have an angle in gradians or turns, convert first: 1 turn = 360 degrees = 2pi radians, and 1 gradian = 0.9 degrees.
Common sectors and their properties
| Name | Angle | Fraction | Sector area | Arc length | Chord length |
|---|---|---|---|---|---|
| 30° sector | 30° (π/6) | 1/12 | πr²/12 | πr/6 | r |
| 45° sector | 45° (π/4) | 1/8 | πr²/8 | πr/4 | 0.7654r |
| 60° sector | 60° (π/3) | 1/6 | πr²/6 | πr/3 | r |
| 90° quarter circle | 90° (π/2) | 1/4 | πr²/4 | πr/2 | r√2 |
| 120° sector | 120° (2π/3) | 1/3 | πr²/3 | 2πr/3 | r√3 |
| 180° semicircle | 180° (π) | 1/2 | πr²/2 | πr | 2r |
| 270° sector | 270° (3π/2) | 3/4 | 3πr²/4 | 3πr/2 | r√2 |
| 360° full circle | 360° (2π) | 1 | πr² | 2πr | 0 |
All values are relative to a circle of radius r. Chord length is the straight line joining the arc endpoints.
Frequently asked questions
How do I enter the angle in radians instead of degrees?
Change the "Angle unit" selector to "Radians". The input then expects radians: a full circle is 2pi (about 6.2832), a semicircle is pi (about 3.1416), a quarter circle is pi/2 (about 1.5708). If your angle is in degrees, divide by 57.2958 to convert, or simply leave the selector on "Degrees" and the calculator converts internally.
What is the difference between a sector and a segment?
A sector is the full pie-slice bounded by two radii and the arc. A segment is the smaller region cut off by a single chord, which is the sector with the central isoceles triangle removed. Toggle "Also compute circular segment" and the calculator adds segment area to the results: A-segment = sector area minus half r-squared times sin(theta).
How do I find the radius if I know the sector area and angle?
Set "Solve for" to "Radius (given area + angle)", enter the known sector area and the central angle, and the calculator returns the radius using r = sqrt(2A divided by theta). The arc length, chord length, and segment area are also computed from the solved radius.
How do I find the angle if I know the sector area and radius?
Set "Solve for" to "Angle (given area + radius)", enter the known sector area and radius, and the calculator returns the central angle using theta = 2A divided by r-squared. The result is shown in both degrees and radians.
What length units can I use?
The "Length unit" selector covers inches, feet, yards, millimetres, centimetres, and metres. Areas are automatically reported in the corresponding square unit (for example ft² when feet is chosen). No manual conversion is needed: just pick the unit that matches your measurements.
What if the angle is more than 360 degrees?
A physical sector cannot exceed one full turn, so the angle is treated as at most 2pi radians (360 degrees), which gives the full circle area. Angles larger than that would simply wrap around and add no new area, so they are not meaningful for a single sector.