Angle of Circle Segment Calculator

Calculator Inputs

Known values

Radius

Chord length

Arc length

Segment height

Segment area

Chord segment choice

Length unit label

Decimal precision

Example Data Table

Known Radius	Known Value	Method	Expected Angle	Use Case
10 cm	12 cm chord	Radius and chord	73.739795 degrees	Plate layout
8 m	5 m arc	Radius and arc	35.809862 degrees	Curved track
15 in	4 in height	Radius and height	85.979217 degrees	Pipe fill check
6 ft	20 ft² area	Radius and area	131.136027 degrees	Tank section

Formula Used

The calculator uses radians internally. If radius is r and central angle is θ, then arc length is s = rθ. Chord length is c = 2r sin(θ / 2). Segment height is h = r(1 - cos(θ / 2)). Sector area is 0.5r²θ. Segment area is 0.5r²(θ - sin θ).

When radius and area are entered, the equation cannot be rearranged neatly. The tool solves θ with a bisection method between 0 and 2π radians.

How to Use This Calculator

Select the pair of known values from the first field.
Enter a positive radius.
Enter the matching chord, arc, height, or area value.
Choose minor or major segment when chord length is used.
Add a unit label and choose decimal precision.
Press Calculate to view results above the form.
Use CSV or PDF buttons to export the same result.

Understanding Circle Segment Angles

A circular segment appears when a chord cuts a circle. The curved boundary is the arc. The straight boundary is the chord. The angle usually means the central angle. It is measured at the circle center. This angle controls segment shape. A small angle forms a shallow cap. A larger angle forms a deep segment.

Why This Calculator Helps

Physics tasks often use circular parts. Lenses, tanks, pulleys, pipes, and wave fronts can all involve segments. Manual work can be slow. Each known measurement needs a different equation. This calculator lets you choose the known data. You can use radius with chord, arc length, height, or area. It then returns the angle in radians and degrees. It also estimates chord, arc, height, sector area, triangle area, and segment area.

Practical Measurement Notes

Use one unit system for every length. Do not mix centimeters with inches. The radius must be positive. Chord length cannot be greater than twice the radius. Segment height must be between zero and twice the radius. Arc length should not exceed a full circumference unless you intend extra rotations. Segment area must stay between zero and the full circle area. These checks reduce wrong outputs.

Advanced Result Reading

The calculator reports the central angle first. It also shows the minor or major nature of the segment. A result below one hundred eighty degrees is a minor segment. A result above one hundred eighty degrees is a major segment. The sector area is the pizza slice area. The triangle area is the isosceles triangle inside that sector. The segment area equals sector area minus triangle area. For major segments, the same formula still works because sine becomes negative after one hundred eighty degrees.

Use Cases

Engineers can estimate curved plate geometry. Students can verify homework. Designers can compare arc shapes. Teachers can create examples quickly. The export tools help save calculations. The CSV file works well for spreadsheets. The PDF file gives a simple printable summary. Together, these outputs make the calculator useful for reports, worksheets, and lab notes. Always review assumptions before using results in safety critical design. When uncertainty exists, measure again. Compare inputs before final decisions are made with field notes.

FAQs

What is the angle of a circle segment?

It is usually the central angle formed by two radii drawn to the chord endpoints. This calculator reports that angle in radians and degrees.

Can I calculate a major segment?

Yes. Choose the major segment option when using chord length. Area, height, and arc inputs already define the matching angle directly.

Why is radius required?

The radius connects every segment measure to the circle center. Chord, arc, height, and area formulas all need radius for accurate angle calculation.

What units should I use?

Use one consistent length unit. If radius is in centimeters, chord, arc, and height should also use centimeters. Area then uses square centimeters.

Why can chord length give two answers?

The same chord can create a small cap or a large remaining segment. The calculator lets you choose minor or major segment for this reason.

How is area solved?

The segment area equation includes θ and sin θ. It has no simple direct rearrangement, so the calculator uses a reliable bisection search.

Can this help with physics problems?

Yes. Circular segments appear in optics, tanks, pulleys, pipes, and circular motion diagrams. The derived values help model curved geometry.

What do the export buttons do?

The CSV button creates spreadsheet-ready results. The PDF button creates a simple printable summary with the main calculated values.