Circle Segment Area
Calculate the area, arc length, chord length, and segment height of a circle segment from radius and central angle. Uses A = ½r²(θ-sin θ) with θ in radians.
Circle Segment — Area, Arc, Chord, Height
A = ½r²(θ − sin θ). Enter radius and central angle (degrees).
📐 Real-World Examples — Click to Load
Inputs
⚠️For educational and informational purposes only. Verify with a qualified professional.
📋 Key Takeaways
- • A circle segment is the region bounded by a chord and its arc — like a pizza slice without the tip
- • The area formula A = ½r²(θ − sin θ) requires θ in radians (multiply degrees by π/180)
- • Arc length = rθ, chord = 2r·sin(θ/2), height = r(1 − cos(θ/2))
- • A semicircle (θ = 180°) has area πr²/2 — exactly half the full circle
- • Segments appear in arches, pizza slices, moon phases, and fountain basins
💡 Did You Know?
📖 Formulas Explained
Segment Area
The area equals the sector area minus the isosceles triangle: A = ½r²θ − ½r²sin θ = ½r²(θ − sin θ).
Arc Length
Chord Length
Segment Height
🎯 Expert Tips
💡 Angle in Radians
Always convert degrees to radians: θ_rad = θ_deg × π/180. A 60° angle = π/3 ≈ 1.047 rad.
💡 Pizza Slicing
For equal slices, divide 360° by the number of slices. 6 slices = 60° each; 8 slices = 45° each.
💡 Semicircle Check
When θ = 180°, sin θ = 0, so A = ½r²π = πr²/2 — the semicircle area.
💡 Small Angles
For very small θ, sin θ ≈ θ, so A ≈ ½r²(θ − θ) ≈ 0. The segment becomes negligible.
⚖️ Comparison Table
| Feature | This Calculator | Basic | Manual |
|---|---|---|---|
| Segment area, arc, chord, height | ✅ | ❌ | ⚠️ |
| Step-by-step solutions | ✅ | ❌ | ✅ |
| Charts & visualization | ✅ | ❌ | ❌ |
| Copy & share | ✅ | ❌ | ❌ |
| 7 real-world examples | ✅ | ❌ | ❌ |
| Unit support | ✅ | ⚠️ | ❌ |
📊 Quick Facts
❓ FAQ
What is a circle segment?
A segment is the region bounded by a chord and the arc it subtends. Unlike a sector (pizza slice), it does not include the center.
Why must θ be in radians?
The derivative of sin θ with respect to θ (in radians) gives cos θ. The formula A = ½r²(θ − sin θ) is derived using calculus with radians.
How do I convert degrees to radians?
Multiply by π/180. Example: 60° = 60 × π/180 = π/3 ≈ 1.047 rad.
What is the area of a semicircle?
For θ = 180° = π rad, sin π = 0, so A = ½r²π = πr²/2 — half the circle.
Can the segment be larger than a semicircle?
Yes. For θ > 180°, you get the major segment. The formula still applies; sin θ is negative, so (θ − sin θ) increases.
How is chord length related to the angle?
Chord c = 2r·sin(θ/2). For a diameter (θ = 180°), c = 2r. For small θ, c ≈ rθ.
Where are segments used in real life?
Pizza slices, arch windows, bridge arches, fountain basins, moon phases, dartboard sections.
What is segment height?
The perpendicular distance from the chord to the arc at its midpoint. h = r(1 − cos(θ/2)).
📚 Sources
⚠️ Disclaimer: Results are based on Euclidean geometry. Real-world measurements may vary. For construction and engineering, verify with a professional.