Calculator
Plotly graph
This graph uses radius 10 and shows how arc length, chord length, and sector area change as the central angle increases.
Example data table
| Radius | Central Angle (°) | Arc Length | Chord Length | Sector Area | Inscribed Angle (°) |
|---|---|---|---|---|---|
| 10 | 15 | 2.618 | 2.6105 | 13.09 | 7.5 |
| 10 | 30 | 5.236 | 5.1764 | 26.1799 | 15 |
| 10 | 45 | 7.854 | 7.6537 | 39.2699 | 22.5 |
| 10 | 60 | 10.472 | 10 | 52.3599 | 30 |
| 10 | 90 | 15.708 | 14.1421 | 78.5398 | 45 |
| 10 | 120 | 20.944 | 17.3205 | 104.7198 | 60 |
Formula used
Central angle from arc length: θ = s / r
Central angle from sector area: θ = 2A / r²
Central angle from chord: θ = 2 × asin(c / 2r)
Arc length from angle: s = rθ
Sector area from angle: A = ½r²θ
Chord length from angle: c = 2r sin(θ / 2)
Inscribed angle relation: Inscribed angle = Central angle / 2
Fraction of circle: Percentage = (θ° / 360) × 100
Use radians in the main formulas. When inputs are in degrees, convert first: radians = degrees × π / 180.
How to use this calculator
- Select the calculation mode that matches your known values.
- Enter the radius of the circle.
- Provide the matching input value such as arc length, sector area, chord length, central angle, or inscribed angle.
- Choose degrees or radians when the selected mode needs an angle unit.
- Pick the number of decimal places you want in the output.
- Press Calculate to show the result block above the form.
- Review the graph and comparison values for better interpretation.
- Use the CSV or PDF buttons to export the current result.
About this angle circle calculator
This page is designed for quick circle geometry work where one value leads to many others. A single entry can produce central angle, inscribed angle, arc length, chord length, sector area, circle fraction, full circumference, and full area. That makes the page useful for study, drafting checks, classroom practice, and technical review.
Different circle problems begin with different known values. Some questions give the arc length and radius. Others provide chord length, sector area, or an inscribed angle. Instead of switching between formulas manually, this calculator groups those paths into one form and returns linked outputs together. That helps reduce repeated conversions and common algebra mistakes.
The graph adds another layer of understanding. As the central angle increases, arc length grows linearly, while chord length and sector area change at different rates. Seeing that relationship makes it easier to compare values and verify whether an answer looks reasonable before using it elsewhere.
The export tools support documentation and reuse. CSV works well for spreadsheets and quick records. PDF is useful when you want a simple shareable snapshot of the result. The example table and formula section are included so the page remains practical even before any custom input is entered.
FAQs
1. What does this calculator solve?
It solves circle angle relationships using radius with arc length, sector area, chord length, central angle, or inscribed angle. It also returns related circle measures in one step.
2. Why does the calculator use radians in formulas?
Arc length and sector area formulas work directly with radians. If you enter degrees, the page converts them first, then calculates the remaining values accurately.
3. What is the difference between central and inscribed angles?
A central angle has its vertex at the circle center. An inscribed angle has its vertex on the circle. For the same intercepted arc, the inscribed angle is half the central angle.
4. Can the chord be longer than the diameter?
No. The diameter is the longest possible chord in a circle. If your chord value exceeds twice the radius, the input is not valid for a real circle.
5. Why is my arc length limited to one circumference?
This calculator treats one sector of one circle. Because of that, the arc length cannot be larger than the full circumference for a single complete turn.
6. When should I use the sector area mode?
Use it when the problem gives radius and the area of the slice-shaped region. The calculator converts that area into the central angle, then finds the linked measures.
7. What does the graph help me understand?
The graph shows how arc length, chord length, and sector area change as the central angle grows. It is useful for comparison, trend checking, and quick visual validation.
8. Can I save the result for reports or homework?
Yes. After calculation, you can export the shown result as CSV or PDF. That makes it easier to store, share, or attach to supporting notes.