Calculator form
Use one method at a time. The page returns the angle and all major circle measures together.
Example data table
These sample values show how the calculator behaves across common circle relationships.
| Method | Known size | Known value | Expected angle | Quick note |
|---|---|---|---|---|
| Arc length + radius | r = 10 cm | s = 15.708 cm | 90° | Quarter-circle arc. |
| Chord + radius | r = 12 cm | c = 12 cm | 60° | Minor angle solution. |
| Sector area + radius | r = 8 cm | A = 50.265 cm² | 90° | Area of one quarter sector. |
| Fraction of circle | d = 30 cm | 25% | 90° | One fourth of a full turn. |
| Direct angle | r = 5 cm | 2.094 rad | 120° | Useful for derived measures. |
Formula used
Use this when arc length s and radius r are known. The result is in radians.
Use this when chord length c and radius r are known. The minor angle is produced first.
Use this when sector area A and radius r are known. The result is in radians.
Use this when the sector is a fraction f of the full circle. For percentages, convert first.
Derived measures
Arc length: s = rθ
Chord length: c = 2r sin(θ / 2)
Sector area: A = ½r²θ
Segment area: Asegment = ½r²(θ − sinθ)
Sector perimeter: P = 2r + s
How to use this calculator
- Choose the method that matches your known circle data.
- Select whether your size input is a radius or a diameter.
- Enter the size value and the known arc, chord, area, fraction, or angle.
- Pick the output units you want for linear and area results.
- For chord input, decide whether you want the minor or major central angle.
- Click the button to calculate the angle and all related measures.
- Review the details table, graph, and formula summary.
- Export the computed values as CSV or PDF when needed.
Frequently asked questions
1) What is a central angle?
A central angle is an angle whose vertex sits at the circle’s center. Its sides are radii, and it determines the size of the intercepted arc and sector.
2) Why does the arc formula return radians first?
The arc-length relationship is naturally defined as s = rθ when θ is in radians. The calculator converts that value to degrees after computing the primary result.
3) Why can one chord produce two angles?
A chord can belong to both a minor sector and a major sector. Both share the same endpoints, so the calculator lets you choose the minor or major central angle.
4) Can I use diameter instead of radius?
Yes. Select diameter in the form. The page converts diameter to radius internally, then computes the angle and every related circle measure automatically.
5) What is the difference between sector area and segment area?
Sector area includes the wedge formed by two radii and the arc. Segment area is only the curved region between the chord and the intercepted arc.
6) What happens if my values exceed a full circle?
The calculator checks for impossible or oversized values. It shows an error when the resulting central angle would exceed one full revolution.
7) Is the graph based on my current radius or diameter?
The graph uses the current circle size after converting any entered diameter into a radius. This keeps the plotted arc, chord, and area values consistent.
8) What do the CSV and PDF files contain?
Both exports summarize the computed angle, circle dimensions, perimeter, areas, and percentage share. They are useful for reports, class notes, and design checks.