You know the basics: A = πr² and C = 2πr. But circles have more — arcs, sectors, segments, and chords. This guide covers everything from basic area to advanced circle geometry.
Basic Circle Properties
| Property | Formula | If r = 5 | |----------|---------|----------| | Diameter | d = 2r | 10 | | Circumference | C = 2πr | 31.42 | | Area | A = πr² | 78.54 |
Arc Length
An arc is a portion of the circumference. Its length depends on the central angle θ (in degrees):
Arc length = (θ/360) × 2πr
Example: Arc with θ = 60°, r = 7
Arc = (60/360) × 2π(7) = (1/6) × 43.98 = 7.33 units
Example: Arc with θ = 120°, r = 10
Arc = (120/360) × 2π(10) = (1/3) × 62.83 = 20.94 units
Sector Area
A sector is a "pie slice" of the circle. Its area:
Sector area = (θ/360) × πr²
Example: 90° sector with r = 8
A = (90/360) × π(64) = (1/4) × 201.06 = 50.27 sq units
Example: 45° sector with r = 12
A = (45/360) × π(144) = (1/8) × 452.39 = 56.55 sq units
Segment Area
A segment is the area between a chord and the arc. It's the sector minus the triangle:
Segment = Sector − Triangle
Segment area = (θ/360)πr² − (1/2)r²sin(θ)
Example: 60° segment with r = 10
- Sector = (60/360)π(100) = 52.36
- Triangle = (1/2)(100)sin(60°) = 50 × 0.866 = 43.30
- Segment = 52.36 − 43.30 = 9.06 sq units
Chord Length
A chord is a straight line connecting two points on the circle:
Chord length = 2r × sin(θ/2)
Example: Chord for 90° arc, r = 6
Chord = 2(6) × sin(45°) = 12 × 0.707 = 8.49 units
Complete Formula Reference
| Property | Formula | |----------|---------| | Radius | r | | Diameter | d = 2r | | Circumference | C = 2πr = πd | | Area | A = πr² | | Arc length | L = (θ/360) × 2πr | | Sector area | A = (θ/360) × πr² | | Segment area | (θ/360)πr² − ½r²sinθ | | Chord length | 2r sin(θ/2) |
Real-World Applications
| Application | What to Calculate | |-------------|------------------| | Pizza size comparison | Area = πr² | | Fence around circular garden | Circumference = 2πr | | Sprinkler coverage area | Area = πr² | | Curved road length | Arc length | | Pie chart segment | Sector area | | Tunnel cross-section | Area + segment |
The Trench Truth: When comparing circular areas, small radius changes have a BIG impact because area scales with r². A 10% increase in radius gives a 21% increase in area. A 14" pizza isn't 17% bigger than a 12" — it's 36% bigger. Always compare areas, not diameters.
Related: Area Converter · Square Root Calculator · Quadratic Formula Calculator
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