The sector area calculator works out the area of a slice of a circle — the wedge bounded by two radii and the arc between them. It is a specific job: this page is about a part of a circle defined by an angle, not the whole circle. For the complete circle, including area, circumference and diameter relationships, use our circle calculator instead.
Arb Digital built this one because the sector calculation has a companion that people constantly confuse it with. A sector is the whole wedge, crust included. A segment is the smaller region cut off by the straight chord, with the triangle removed. This tool gives both, so you never have to guess which one your problem meant.
What This Sector Area Calculator Does
Enter a radius and a central angle and it returns the sector area as the headline figure, plus the arc length along the curved edge, the chord length across the open end, the area of the circular segment, and the full perimeter of the sector.
A radius of 10 cm and an angle of 60° gives a sector area of about 52.36 cm², an arc length of 10.472 cm, a chord of exactly 10 cm, a segment area of 9.059 cm² and a sector perimeter of 30.472 cm. The angle can be entered in degrees, radians, as a percentage of the full circle, or in turns — which is useful when your starting point is a proportion rather than an angle, as with a pie chart.
How to Use It
- Enter the radius, measured from the centre point to the curved edge. If you only have the diameter, halve it first.
- Enter the central angle and pick its unit. If you know the slice as a fraction — a quarter, 15% of a chart — use the percentage or turns option and skip the conversion.
- Read the sector area for the whole wedge, or the segment area if your shape is bounded by the straight chord rather than by the two radii.
- Use the arc length for anything running along the curved edge, and the chord for the straight distance across the opening.
- Check the sector perimeter if you need edging all the way round — it is the arc plus two radii, not the arc alone.
The Formula: How Sector Area Is Calculated
In radians the formula is compact: A = ½ r²θ. With r = 10 and θ = 60° = π/3 ≈ 1.047198 radians, A = ½ × 100 × 1.047198 = 52.36 cm².
In degrees it is the proportional form: A = (θ / 360) × πr². The same example gives (60/360) × π × 100 = (1/6) × 314.159 = 52.36 cm². Both routes agree exactly, because they are the same statement — a sector's area is the circle's area scaled by the fraction of the full turn the angle covers. Which form is more convenient depends only on the unit your angle arrived in.
The other three figures follow from the same angle. Arc length is s = rθ in radians, giving 10.472 cm here. Chord length is c = 2r sin(θ/2) = 2 × 10 × sin 30° = 10 cm, which is exactly the radius — a 60° chord always equals the radius, because the triangle formed is equilateral. MathWorld's circular sector entry sets out the formal derivations.
Sector or Segment? They Are Not the Same Region
This is the distinction that causes most of the errors. A sector is the pizza-slice region bounded by two radii and the arc: point at the centre, crust at the edge. A segment is the region between the chord and the arc: the piece you would cut off with a single straight knife stroke that never passes through the centre.
The relationship between them is a triangle. The sector equals the segment plus the isosceles triangle formed by the two radii and the chord. So the segment area is A = ½ r²(θ − sin θ), with θ in radians. For our example that is ½ × 100 × (1.047198 − 0.866025) = 50 × 0.181173 = 9.059 cm², against a sector area of 52.36 cm². The segment is barely a sixth of the sector, and if you needed one and calculated the other you are out by a factor of nearly six. MathWorld's circular segment entry lists the full set of segment relations, including the forms based on segment height rather than angle.
Which one you need depends on the physical problem. The fill area in a partly filled horizontal cylindrical tank is a segment, not a sector — the liquid surface is a flat chord. The coverage of a rotating sprinkler or a security camera's field of view is a sector, since it sweeps out from a central point. Getting this wrong is the most expensive mistake on this page, which is why both figures are shown together.
Radians, Degrees and Why the Formula Is Simpler in Radians
A radian is the angle at which the arc length equals the radius. There are 2π ≈ 6.28319 radians in a full turn, so one radian is about 57.2958°. Radians are not an arbitrary alternative to degrees; they are the unit that makes circular formulas lose their conversion constants.
Compare the two forms. In radians, arc length is s = rθ and sector area is A = ½r²θ. In degrees, the same quantities are s = 2πr(θ/360) and A = πr²(θ/360), each carrying a conversion factor. Degrees are a human convention — 360 is convenient because it divides so many ways — while radians are the ratio the geometry actually uses. MathWorld's radian entry gives the formal definition and the reasons the unit is preferred in analysis.
The practical rule is to convert to radians before applying any circular formula, then convert back at the end if your audience expects degrees. This calculator does that internally, which is why you can enter degrees, radians, percentages or turns and get consistent answers. Our angle converter handles the conversion on its own if you need it elsewhere.
Sectors and Pie Charts
A pie chart is a set of sectors, and the conversion between a percentage and an angle is 3.6° per percentage point, since 360 ÷ 100 = 3.6. A 25% share occupies 90°, a 15% share 54°. Enter the percentage directly in this calculator to get the corresponding area without converting.
There is a design point buried in that arithmetic. The area of a sector is proportional to its angle at a fixed radius, so a pie chart does encode its data honestly by area. The problem is that people are considerably better at comparing lengths than angles or areas, which is why closely sized slices are hard to rank by eye. The same numbers in a bar chart are read accurately in a moment. If you are choosing a chart type rather than calculating one, that is a reason to prefer bars for anything beyond a handful of categories.
Where Sector Calculations Are Used
Rotating irrigation sprinklers cover sectors, and specifying one means matching its radius and arc setting to the area needing water. A 10 m sprinkler set to a 90° arc covers ¼ × π × 100 = 78.5 m². Set the same head to 180° and it covers twice that, but the flow rate stays fixed, so the application depth per hour halves — a genuine and often missed consequence of changing an arc setting.
Fan-shaped garden beds, curved patio corners, radial paving, quadrant shower trays, curved seating and fabric gores are all sector calculations. So is the swept area of a windscreen wiper, the coverage cone of a camera or sensor, and the material needed to form a cone — a cone unrolls flat into a sector, whose radius is the cone's slant height and whose arc length equals the base circumference.
For the curved edge on its own, our arc length calculator covers that specifically, including working backwards from a measured arc to the angle. For the full circle, the circle calculator is the right page.
Arb Digital builds calculators that answer one question properly, rank for the searches behind it, and put your business in front of people already looking. Browse the full set, or talk to us about tools for your own audience.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Calculating a sector when you needed a segment — the segment excludes the central triangle and can be a small fraction of the sector's area.
- Using degrees in the radian formula — putting 60 into A = ½r²θ instead of 1.047198 overstates the area by a factor of about 57.
- Entering the diameter as the radius — that quadruples the area, because area depends on the radius squared.
- Treating the arc length as the sector's perimeter — the perimeter is the arc plus the two straight radii.
- Assuming an angle over 360° means a bigger area — beyond a full turn the sector simply overlaps itself, so this calculator caps the input at one complete revolution.
Related Free Tools From Arb Digital
Measure the curved edge with the arc length calculator, work out the whole circle with the circle calculator, convert between angle units with the angle converter, handle straight-sided shapes with the polygon calculator or the rectangle area calculator, and convert a finished area with the area converter. Everything is listed in the free online tools hub.
Frequently Asked Questions
In radians it is half the radius squared multiplied by the angle: A = ½r²θ. In degrees it is the angle divided by 360, multiplied by the full circle area πr². A 60° sector of a 10 cm circle has an area of about 52.36 cm².
A sector is the wedge bounded by two radii and the arc, like a pizza slice. A segment is the smaller region between a straight chord and the arc, with the central triangle removed. The sector equals the segment plus that triangle.
Use A = ½r²(θ − sin θ), with the angle in radians. For a 10 cm radius and a 60° angle that gives about 9.06 cm², compared with a sector area of 52.36 cm².
Multiply the percentage by 3.6 to get degrees, since a full circle of 360° represents 100%. A 25% share is 90°. This calculator accepts percentages directly.
It is the arc length plus two radii, because the wedge is bounded by the curved edge and the two straight sides. For a 10 cm radius at 60° that is 10.472 + 20 = 30.472 cm.
Because the two radii and the chord form a triangle with a 60° apex and two equal sides, which makes all three angles 60°. An equilateral triangle has all sides equal, so the chord matches the radius.
Not meaningfully. Beyond one full revolution the sector overlaps itself and the area stops increasing, so this calculator limits the angle to a single complete turn.
