Circular Segment Area Calculator: Formula & Examples

A circular segment area calculator finds the area of the region between a chord and the arc of a circle. You enter the circle’s radius and either the central angle or the segment height, and the calculator returns the segment’s area in square units. This is useful for geometry problems, arch designs, and any project involving a sliced-off portion of a circle.

Formula

The standard formula for the area of a circular segment (the region bounded by a chord and the arc it cuts off) is:

Area = ½ × r² × (θ – sin θ)

Where:

• r = radius of the circle (any unit of length)
• θ = central angle in radians (the angle subtended by the chord at the center)

If your angle is in degrees, convert it first: θ (radians) = θ (degrees) × π / 180. The calculator above does this conversion automatically. The result is in square units of whatever length unit you used for the radius (e.g., square feet if radius is in feet, square meters if radius is in meters).

Worked Example

Using the default values from the calculator: radius = 10 ft, central angle = 60°.

1. Convert the angle to radians: 60° × π / 180 = 1.0472 radians.
2. Compute sin(θ): sin(1.0472) ≈ 0.8660.
3. Plug into the formula: Area = ½ × (10)² × (1.0472 – 0.8660) = ½ × 100 × 0.1812 = 9.06 ft².

The calculator shows 9.06 ft², matching the manual calculation. The chord length for this segment is 2 × 10 × sin(30°) = 10 ft, and the segment height is 10 × (1 – cos(30°)) = 1.34 ft.

When to Use This Calculator

You need a circular segment area when a problem involves a chord cutting across a circle, not a full sector. Common situations include:

• Calculating the area of a circular arch or a lens-shaped window.
• Finding the cross-sectional area of a partially filled horizontal cylindrical tank (e.g., a fuel tank lying on its side).
• Designing a garden bed that follows a circular border with a straight edge.
• Solving geometry homework or exam problems.

If you instead need the area of the whole sector (the pie-slice shape), use the sector area formula: Area = ½ × r² × θ. The segment is that sector minus the triangular part below the chord.

How to Read the Result

The output is the area of the segment only—the part between the chord and the arc. It is always positive and less than the area of the full circle. For a given radius, the segment area increases as the central angle increases, up to a maximum when the angle is 360° (the whole circle). At very small angles, the segment area approaches zero, and the chord length approaches the arc length.

The calculator also gives the chord length and the segment height (the maximum distance from the chord to the arc). These are useful when you need to lay out the shape physically, such as marking an arch on a wall or cutting a curved piece of material.

Common Mistakes and How to Avoid Them

• Using degrees in the formula directly: The formula requires radians. If you enter degrees without converting, you’ll get a wrong (often negative) result. The calculator handles this, but if you do it by hand, remember to multiply by Ï€/180.
• Confusing segment with sector: A sector is the whole pie slice; a segment is only the curved part. Many people accidentally use the sector formula when they need the segment.
• Mixing units: If the radius is in feet, the area is in square feet. Don’t mix meters and centimeters without converting.
• Using the height instead of the angle: If you only know the segment height (h), you can find the angle via θ = 2 × arccos(1 – h/r). But it’s easy to make a mistake; this calculator expects the angle directly.

Limitations and Alternatives

This formula assumes a perfect circle and a straight chord. Real-world objects may have irregularities, so the result is an approximation for non-perfect shapes. Also, the angle must be between 0° and 360°; the calculator enforces that range.

If you know the radius and the segment height instead of the angle, you can still compute the area using a different formula: Area = r² × arccos((r – h)/r) – (r – h) × √(2rh – h²). Many online calculators offer that option, but this one focuses on the angle input because it’s the most common in geometry problems.

For a partially filled tank, the segment area times the tank length gives the liquid volume. That’s a separate calculation, but this area is the first step.

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 between them—like a slice of pie. A segment is bounded by a chord and the arc, so it’s the sector minus the triangle formed by the two radii and the chord. In short, a segment has a straight edge (the chord), while a sector has two straight edges (the radii).

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

Yes. From the chord length (c) and radius (r), you can find the central angle using θ = 2 × arcsin(c / (2r)). Then use the same formula. If you don’t want to do that step, you can use a calculator that accepts chord length directly, but you’ll need to convert first.

What if my central angle is more than 180°?

The formula still works for angles up to 360°. For angles between 180° and 360°, the segment becomes the larger part of the circle (the major segment). The chord is still straight, but the arc is the long way around. The area will be larger than a semicircle, up to the full circle at 360°.

Why does the calculator show a negative area if I enter an angle above 180°?

That shouldn’t happen with this calculator because it clamps the angle to 0–360°. However, if you use the formula manually with an angle in degrees without converting to radians, sin(θ) can be negative for angles above 180°, making the term (θ – sin θ) negative. Always convert to radians first.

How do I find the area of a segment if I only know the height and radius?

Use the alternative formula: Area = r² × arccos((r – h)/r) – (r – h) × √(2rh – h²). This is derived from the same geometry, but it’s easier to use a dedicated calculator for that input. If you have the height, you can also compute the angle as θ = 2 × arccos(1 – h/r) and then use the standard formula.

Related

• Area Calculator for Feet and Inches
• Area Calculator UK: Measure Square Metres and Feet
• Acre-Feet to Gallons Calculator and Conversion Formula

in r | Radius | ft | 10 | 0 | 100000
in a | Central angle | deg | 60 | 0 | 360
out area | Segment area | ft² | 0.5 * r^2 * (a * pi / 180 - sin(a * pi / 180)) | 2
out chord | Chord length | ft | 2 * r * sin(a * pi / 360) | 2
out h | Segment height | ft | r * (1 - cos(a * pi / 360)) | 2