How to use the Sector Area Calculator
- Enter the circle's radius.
- Enter the central angle of the sector in degrees.
- Read the sector area — the full pie slice including the centre.
- Read the segment area — just the region between the chord and the arc.
- Use the percentage row to check the sector against the whole circle.
How the calculation works
A sector is a fraction of a circle, and the fraction is the central angle divided by the full turn. Multiplying that fraction by πr² gives the sector area; in radians it simplifies to half r squared times the angle. This is the calculation behind every pie chart, every fan-shaped irrigation zone and every angular field of view.
The segment is what remains after removing the triangle formed by the two radii and the chord. Since that triangle's area is half r squared times the sine of the angle, the segment is half r squared times the angle minus its sine. It is the shape of liquid in a partly filled horizontal cylinder, which makes it the key to dipstick calibration and tank ullage tables.
That tank application is where the segment formula earns its keep, and where intuition fails badly. A horizontal cylinder filled to a quarter of its diameter is not a quarter full — it holds only about 19.6% of capacity, because the narrow bottom of the circle contributes far less volume than the wide middle. Anyone reading a horizontal tank by depth without a proper strapping table will consistently misjudge stock.
sector = ½r²θ; segment = ½r²(θ − sin θ), with θ in radiansSource: Circular sector and segment formulas; segment method used in API MPMS tank calibration practice.
Worked example
A horizontal cylindrical tank of 1.2 m radius is filled to a depth of 0.6 m. What fraction is in the tank?
- Half-depth means the fill surface passes through the centre, so the wetted area is exactly half the circle.
- Half circle area: π × 1.2² / 2 = 2.262 m².
- By contrast a 0.3 m depth gives a 120° segment: ½ × 1.44 × (2.0944 − 0.8660) = 0.884 m².
- That is 0.884 / 4.524 = 19.5% of the cross-section, for 25% of the depth.
Half depth is exactly half full, but quarter depth is only 19.5% full — the non-linearity that makes strapping tables necessary.
Frequently asked questions
What is the difference between a sector and a segment?+
A sector includes the centre and both radii, like a pizza slice. A segment is only the part beyond the chord, like the crust end of that slice.
How do I get the segment area from a fill depth?+
Convert depth to the central angle first: θ = 2·arccos((r − h)/r), then apply the segment formula.
Can the angle exceed 180 degrees?+
Yes. A reflex sector is simply the rest of the circle, and the formula handles it — the segment then exceeds the semicircle.
Where else does the segment formula appear?+
Partly filled pipes in drainage hydraulics, the overlap area of two circles, and the lens shapes in Venn-style area calculations.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.