Segment Area Calculator - Area, Chord, and Central Angle

A segment area calculator finds the area of a circular segment, which is the region between a chord and the arc it cuts off from a circle. Instead of measuring the full circle, the segment area calculator gives you the smaller, curved piece that the chord and the arc form together, which is the shape that comes up whenever a flat edge meets a curve.

• • Engineering and fabrication: Estimate the area of pipe cross-sections, lens cutouts, and tank end-caps where a flat face meets a cylindrical curve.
• • Design and architecture: Plan arched windows, doorway reveals, ornamental mouldings, and circular planters where the curved and flat surfaces meet.
• • Surveying and land work: Calculate the area of curved plots, pond edges, or paving segments where one boundary is a straight chord.
• • Classroom and homework problems: Solve textbook and exam questions on circular segments, sagitta, and chord length without re-deriving the formula each time.

The shape shows up whenever something curves back to a straight line, which is most of the time you measure a partial circle in real life. A circular segment is not the same as a sector: the sector is the pie-slice bounded by two radii, while the segment is bounded by a chord on one side and the arc on the other. That distinction matters because the sector-segment relationship flips: the sector is the larger piece for a minor segment, and the segment is the larger piece for a major segment.

Working with this calculator means you only need two numbers: the radius of the parent circle and the segment height, also called the sagitta. The tool returns the area plus the chord length, the central angle in degrees and radians, and the sector area, so you can sanity-check the answer without switching tools.

If you need the full circle instead of a curved piece, the Circle Calculator covers the diameter, circumference, and area of the whole shape in one place.

The calculator starts with the radius and the height of the segment, then converts those into a central angle and the segment area through standard circle geometry.

• r: Radius of the parent circle, the distance from the center to the edge in the chosen length unit.
• h: Segment height or sagitta, the perpendicular distance from the chord to the arc. Valid range is 0 to 2r.
• theta: Central angle in radians that subtends the segment, derived as 2 arccos((r - h) / r).

The height-based form of the formula is convenient when you know the sagitta but not the angle, which is the usual case for a measurement taken from a drawing. The trig form, (1/2) r^2 (theta - sin theta), is convenient when you already know the central angle and just need the area.

Both forms give the same number. Internally, the calculator finds the half-angle from (r - h) / r, doubles it for the central angle, then applies (1/2) r^2 (theta - sin theta) to compute the area, chord, and sector area in one pass.

According to Wikipedia: Circular segment, the area of a circular segment is A = (1/2) r^2 (theta - sin theta), where r is the radius and theta is the central angle in radians that the segment subtends.

If you also need the curved boundary of the segment, the Arc Length Calculator converts the same central angle into the arc length and the matching sector area in one step.

Four ideas cover most of what you need to interpret a circular segment result: the parent circle, the radius, the sagitta, and the sector that the segment is part of.

Keeping the segment height in mind avoids the common trap of treating the segment as the sector. The sector uses two radii as boundaries, while the segment uses one chord, and which one is larger depends on whether the sagitta h is below or above the radius r.

If you want to subtract the chord triangle from the sector to recover the segment by hand, the Triangle Calculator covers the side-angle-side and base-height setups you would use for that triangle.

The inputs are the two measurements that describe a circular segment. After typing them in, the calculator returns the segment area and the supporting values in real time.

1. 1 Enter the radius: Type the radius of the parent circle. Use the radius, not the diameter, since the segment formulas rely on r directly.
2. 2 Enter the segment height: Type the sagitta, which is the distance from the chord to the arc. The valid range is 0 to 2r, with h = r producing a semicircle.
3. 3 Choose the length unit: Pick the unit that matches your drawing or measurement so the chord, area, and sector area are reported in the same system.
4. 4 Read the segment area: The primary result is the segment area in square units, with the unit shown next to the value so you can quote it directly.
5. 5 Cross-check with chord and angle: Use the chord length and central angle to verify the result against any other measurement you took from the shape.

A 10 cm radius and a 4 cm sagitta give a segment area of 44.7295 cm^2, a chord of 16 cm, and a central angle of 106.26 degrees. If your drawing also shows a chord of 16 cm, the result is consistent and you can quote the area with confidence.

For layouts where the curved piece is one of many regions, the Area Calculator helps you add segment areas to rectangles, triangles, and other regular shapes without re-entering units.

A focused segment area calculator saves time on problems that mix circle geometry with practical measurements, and the supporting outputs make the result easier to defend.

• • Direct input from a drawing: Radius and sagitta are the two measurements most often available on a sketch, so you can go from a drawing to a numeric area without first solving for the central angle.
• • Multiple outputs in one step: The chord length, central angle, and sector area come from the same two inputs, which lets you check the result from several angles before committing to a number.
• • Avoids degree-to-radian mistakes: The calculator keeps the trig in radians under the hood, so the usual slip of plugging a degree value into a radian formula is removed from the workflow.
• • Flexible length unit: Centimeters, meters, inches, and feet are all supported, so the area and chord labels match the units on your plan or your measuring tape.
• • Quick extreme-case checks: Inputs of h = 0, h = r, and h = 2r each produce a known result, which is a fast way to confirm the tool is set up correctly before trusting it on a real problem.

These benefits show up most clearly when a project crosses disciplines, such as a survey plan handed to a fabricator or a textbook problem with messy numbers. The calculator keeps the geometry correct while the human focus stays on the part of the problem that needs judgment.

When the curved boundary is part of a larger irregular shape, the Polygon Area Calculator can take the segment area as one piece of a multi-region total without re-measuring anything.

Segment area is sensitive to a small set of inputs, and a few limitations are worth knowing before quoting a number to a client or grading a problem.

• • The calculator assumes a true circular segment. Real shapes such as arches, lens cutouts, and pipe cross-sections may deviate slightly from a true circle, which can shift the area by a few percent.
• • Floating-point rounding is used to keep the displayed values readable, so the area, chord, and angle are rounded to four decimal places. For most practical work this is well below measurement noise.

These caveats match a hand-calculated segment problem. The calculator removes the arithmetic, so the same measurement discipline still applies on the input side.

According to Wolfram MathWorld: Circular Segment, a circular segment's area can be expressed in terms of the radius and sagitta, giving A = r^2 arccos((r - h) / r) - (r - h) sqrt(2 r h - h^2).

When you want to check the chord triangle by hand before subtracting it from the sector, the Scalene Triangle Area Calculator handles the side and base inputs without re-entering the radius.