Circle Segment Calculator: Area, Arc, Height

This calculator works out the area, arc length, chord length, and segment height of a circular segment. You enter the circle’s radius and either the central angle or the segment height, and it returns the full set of geometric properties using standard formulas. The first result shown is the segment area, which is the region between a chord and its arc.

Formula

The segment area is the sector area minus the triangular area between the two radii and the chord. In degrees, the formula is:

A = (π r² / 360) × θ – (r² / 2) × sin(θ)

where:

• A – area of the circular segment (same unit as r², e.g., m²)
• r – radius of the circle (m, ft, cm, etc.)
• θ – central angle in degrees (0 to 360)

If you prefer radians, the formula becomes A = (r² / 2) × (α – sin α), where α is the angle in radians. The calculator converts degrees to radians internally.

Worked Example

Using the default values: radius r = 5 m and central angle θ = 60°.

1. Convert angle to radians: 60° × π/180 = 1.0472 rad.
2. Sector area = (π × 5² / 360) × 60 = (78.5398 / 360) × 60 = 13.0899 m².
3. Triangle area = (5² / 2) × sin(60°) = 12.5 × 0.8660 = 10.8253 m².
4. Segment area = 13.0899 – 10.8253 = 2.2646 m².

The calculator shows 2.2646 m², matching this step-by-step result.

What Is a Circular Segment?

A circular segment is the region bounded by an arc and the chord that connects its endpoints. It looks like a “slice� of a circle with the pointed tip cut off. If the central angle is less than 180°, it is a minor segment; if greater than 180°, it is a major segment (which includes the center). The segment height (also called sagitta) is the maximum distance between the chord and the arc, measured perpendicular to the chord.

This shape appears in many practical places: arches in architecture, the cross-section of a partially filled horizontal tank, the area of a lens, or the portion of a circular garden covered by a sprinkler. Knowing the segment area helps with material estimates, fluid volume calculations, and design geometry.

How to Use the Calculator

Enter the radius and the central angle. The calculator then computes:

• Segment area – the main result, in square units.
• Arc length – the curved boundary length.
• Chord length – the straight line between the arc’s endpoints.
• Segment height – the sagitta, useful for arch design.

All outputs use the same unit you entered for the radius (e.g., if radius is in feet, areas are in square feet and lengths are in feet). If you only know the height and radius, you can solve for the angle using the inverse cosine: θ = 2 × arccos(1 – h/r). Then enter that angle here.

Common Mistakes and Limitations

• Mixing degrees and radians – The calculator expects degrees. If you have radians, multiply by 180/Ï€ to convert.
• Using diameter instead of radius – Always halve the diameter first.
• Confusing segment with sector – A sector is the whole pie slice (arc + two radii). The segment is the sector minus the triangle, so it is always smaller.
• Forgetting the angle unit – The formula with sin(θ) in degrees only works if you convert properly. The calculator does this for you.

The formula assumes a perfect circle and a straight chord. For very large angles (close to 360°), the segment approaches the whole circle area, and the chord length approaches zero. The calculator remains accurate for all angles from 0 to 360 degrees.

Real-World Applications

One common use is determining the volume of liquid in a horizontal cylindrical tank. If the tank has radius r and the liquid depth is h, the cross-sectional area of the liquid is a circular segment with height h. Multiply that area by the tank length to get the volume. This is why engineers and farmers use segment calculators for fuel, water, or chemical storage.

Another use is in architecture for arched windows, doorways, and bridges. The segment height and chord length define the arch’s shape, and the area helps estimate glass or masonry needed. In landscaping, segment areas help compute sod or mulch for circular flower beds with straight borders.

Frequently Asked Questions

What is the difference between a segment and a sector?

A sector is the region enclosed by two radii and the arc between them – it looks like a pizza slice. A segment is the region between the chord and the arc, which is the sector minus the isosceles triangle formed by the two radii and the chord. The segment is always smaller than the sector for the same angle.

Can I use this calculator if I only know the chord length and height?

Yes, but you need to find the radius first. The relation is r = (c² / (8h)) + (h / 2), where c is chord length and h is segment height. Then compute the angle using θ = 2 × arcsin(c / (2r)). Enter those values into the calculator.

How do I find the area of a major segment (greater than 180°)?

The formula works for any angle up to 360°. For angles above 180°, the sine term becomes negative, which effectively adds the triangle area instead of subtracting it, giving the larger segment area. The calculator handles this automatically.

What units should I use?

Use any consistent unit for radius (meters, feet, inches, etc.). The area will be in that unit squared (m², ft², in²), and lengths (arc, chord, height) will be in the same unit as the radius. The angle must be in degrees.

Why is the segment area not proportional to the angle?

Because the triangle area subtracted is not linear with the angle. The sector area is proportional to the angle, but the triangle area depends on sin(θ), which is nonlinear. So doubling the angle does not double the segment area.

Alternatives and Related Calculations

If you need the area of a full circle, use πr². For a sector, use (πr²θ)/360. For an annulus (ring), subtract the inner circle area from the outer. This calculator focuses on the segment, which is the most common non-trivial circular area. For volumes, multiply the segment area by the length of the prism or cylinder.

If you have the arc length instead of the angle, solve for θ = arc length / r (in radians) then convert to degrees. If you have the chord length, use θ = 2 × arcsin(c / (2r)). The calculator is designed to be flexible, but it requires the angle as input.

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in r | Radius | m | 5 | 0
in a | Central angle | deg | 60 | 0
out area | Segment area | m² | (pi * r^2 * (a * pi / 180 - sin(a * pi / 180))) / 2 | 4
out arc | Arc length | m | r * a * pi / 180 | 4
out chord | Chord length | m | 2 * r * sin(a * pi / 180 / 2) | 4
out h | Segment height | m | r * (1 - cos(a * pi / 180 / 2)) | 4