Circle Segment Area Calculator & Formula

This calculator works out the area of a circular segment — the region between a chord and the arc it cuts off — from the circle’s radius and the segment’s height (sagitta). Enter the radius and height in any consistent unit, and you get the segment area, plus the chord length and arc length for reference.

Formula

The standard formula for the area of a circular segment is:

A = r² · cos⁻¹((r − h) / r) − (r − h) · √(2rh − h²)

Where:

• A = area of the segment (same unit squared as the radius)
• r = radius of the circle
• h = height of the segment (the maximum distance from the chord to the arc, also called the sagitta)

The formula uses the inverse cosine (arccos) in radians. The first term gives the area of the circular sector formed by the two radii and the arc; the second term subtracts the triangular area between the chord and the two radii. The result is the area between the chord and the arc.

Worked Example

Using the default values from the calculator above: radius r = 5 m and segment height h = 2 m.

1. Compute (r − h) = 5 − 2 = 3 m
2. Compute (r − h) / r = 3 / 5 = 0.6
3. Compute cos⁻¹(0.6) ≈ 0.927295 radians
4. First term: r² · 0.927295 = 25 · 0.927295 = 23.1824 m²
5. Compute 2rh − h² = 2·5·2 − 2² = 20 − 4 = 16
6. √16 = 4 m
7. Second term: (r − h) · √(2rh − h²) = 3 · 4 = 12 m²
8. Segment area = 23.1824 − 12 = 11.1824 m²

The calculator rounds to 4 decimal places: 11.1824 m².

When to Use a Circle Segment Area Calculator

Circular segment areas appear in many practical fields:

• Civil engineering: volume of a partially filled horizontal cylindrical tank (multiply the segment area by the tank length).
• Agriculture: flow rate in partially filled irrigation pipes or culverts.
• Architecture: area of arched windows, doorways, or decorative wall sections.
• Manufacturing: material needed for a curved cutout or the cross-section of a cam.
• Geometry education: solving textbook problems involving chords and arcs.

If you have the chord length and the radius instead of the height, you can find the height using h = r − √(r² − (c/2)²), then use the same formula. The calculator expects the height directly, so you may need to do that conversion first.

How to Read the Results

The main output is the area of the segment. Two additional values are provided:

• Chord length: the straight-line distance between the two endpoints of the arc. This is the base of the segment.
• Arc length: the curved distance along the circle between the same two points.

These are useful for ordering materials (e.g., trim for the arc) or for checking the geometry. The chord length formula is c = 2√(2rh − h²), and the arc length is s = r · θ, where θ is the central angle in radians (θ = 2·cos⁻¹((r−h)/r)).

All outputs are in the same unit system you used for the inputs. If you enter meters, areas come out in square meters and lengths in meters. If you enter inches, areas are square inches and lengths are inches.

Common Mistakes and Pitfalls

• Unit mix-ups: Mixing meters and centimeters in the same calculation gives nonsense. Convert everything to one unit first.
• Using degrees instead of radians: The formula requires the inverse cosine to be in radians. Most calculators default to radians, but if you use a spreadsheet function like ACOS, it returns radians. If you manually compute with degrees, you must convert.
• Height greater than radius: The segment height cannot exceed the diameter (2r). If h > 2r, the formula returns an error or a nonsensical result. For h = r, the segment is a semicircle (area = πr²/2).
• Confusing segment with sector: A sector is the pie-slice shape between two radii and the arc; a segment is the region between the chord and the arc. The sector area is simply r²·θ/2, which is always larger than the segment area.
• Rounding too early: If you round intermediate values (like the arccos) to 2 decimals, your final answer can be off by several percent. Keep full precision until the final step.

Limitations of the Formula

The formula assumes a perfect circle and a flat chord. In real-world objects, slight irregularities (e.g., a tank that is not perfectly cylindrical) will cause small errors. Also, the formula uses the central angle implicitly; if you have the angle directly, you can use the simpler sector-minus-triangle approach: A = (r²/2)(θ − sin θ), where θ is in radians. That version is equivalent and sometimes easier to remember.

The calculator works for any segment height from 0 (a degenerate segment with zero area) up to the diameter (the whole circle). For heights beyond the radius, the formula still works; the segment simply becomes larger than a semicircle.

Frequently Asked Questions

What is the difference between a segment and a sector?

A sector is the region bounded by two radii and the arc (like a pizza slice). A segment is the region bounded by a chord and the arc — it does not include the center of the circle unless the chord passes through the center (which makes it a semicircle). The segment area is the sector area minus the triangular area between the chord and the two radii.

Can I calculate the segment area if I only know the chord length and the radius?

Yes. First find the height using h = r − √(r² − (c/2)²), where c is the chord length. Then plug that h into the segment area formula. This works for segments smaller than a semicircle; for larger segments, use h = r + √(r² − (c/2)²) because the chord is above the center.

Why does the formula use arccos? What if my calculator gives an error?

The arccos function computes the central angle from the ratio (r−h)/r. If you get an error, check that h is between 0 and 2r. Also ensure that (r−h)/r is between −1 and 1; if h is exactly 2r, the ratio is −1 and arccos(−1) = π, which is fine. If h is slightly larger than 2r, the ratio is less than −1, which is invalid.

How do I convert the segment area to volume for a horizontal tank?

Multiply the segment area by the length of the tank. For example, if the segment area is 11.1824 m² and the tank is 3 m long, the volume is 33.5472 m³. This gives the volume of liquid in a partially filled horizontal cylindrical tank, assuming the liquid level is below the top of the tank.

What is the sagitta?

The sagitta is the height of the segment — the perpendicular distance from the midpoint of the chord to the arc. It is the same as the h used in the formula. In optics and engineering, the sagitta is often used to describe the curvature of a lens or mirror.

Does the formula work for a segment larger than a semicircle?

Yes. The formula is valid for any h between 0 and 2r. When h > r, the segment is more than half the circle, and the arccos term gives an angle greater than π/2, so the sector area is larger than a semicircle. The subtraction of the triangle still yields the correct segment area.

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in r | Radius | m | 5 | 0
in h | Segment height | m | 2 | 0
out area | Segment area | m² | r^2 * acos((r - h) / r) - (r - h) * sqrt(2 * r * h - h^2) | 4
out chord | Chord length | m | 2 * sqrt(2 * r * h - h^2) | 3
out arc | Arc length | m | 2 * r * acos((r - h) / r) | 3