04 // Interactive Formula Compendium // GEOMETRY-TRIGs = r · θ, A_sector = ½ r² θ (θ in radians)
Circular Arc Length, Sector Area & Chord Length Formula Sheet
For a circle of radius r subtending a central angle θ measured in radians, the circular arc length is s = rθ, the pie-slice sector area is A = ½r²θ = ½rs, and the straight-line chord is c = 2r sin(θ/2).
Formula Reference
Main Formula
s = r · θ, A_sector = ½ r² θ (θ in radians)
Rearranged Forms
c = 2r · sin(θ / 2) (Chord Length)
A_segment = ½ r² (θ − sin θ) (Circular Segment Area)
h_sagitta = r · (1 − cos(θ / 2)) (Sagitta / Arc Rise)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| A_sector | Circular Sector Area (½ r² θ) | m² |
| r | Circle Radius (r) | m |
| theta_deg | Central Angle (degrees) | deg |
Inputs & Parameters
[m]
[deg]
Calculated Result (Circular Sector Area (½ r² θ))
A_sector =52.35988m²
Formula Notes & Background
- By definition of radian measure, one radian subtends an arc equal in length to the radius r, so arc length scales linearly as s = rθ.
- A full circle of angle 2π rad has area πr²; taking the fraction θ / (2π) of the full area gives A_sector = (θ / 2π) · πr² = ½r²θ.
- Subtracting the isosceles triangle area ½r² sin(θ) from the sector area ½r²θ leaves the circular segment area ½r²(θ − sin θ).
Step-by-Step Examples
#1: Find the arc length, sector area, and chord for r = 10 m and θ = 60° (π/3 rad).
r = 10 mθ = 60° = π/3 rad ≈ 1.04720 rad
s = 10 × (π/3) = 10.472 m; A = ½ × 100 × (π/3) = 52.360 m²; c = 20 sin(30°) = 10 m
→ Arc s = 10.472 m, Sector Area = 52.360 m², Chord = 10.000 m
#2: Find the circular segment area bounded by the chord and arc in the example above.
r = 10 mθ = π/3 rad
A_seg = ½(100)(π/3 − sin(π/3)) = 50(1.04720 − 0.86603)
→ A_segment = 9.059 m²
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Frequently Asked Questions
How do you convert degrees to radians before using s = rθ?+
Multiply the angle in degrees by π / 180 (e.g., 90° × π/180 = π/2 rad ≈ 1.5708 rad).
What is the sagitta of an arc and where is it used?+
The sagitta h = r(1 − cos(θ/2)) is the perpendicular height from the midpoint of the chord to the peak of the arc, widely used in optical lens grinding and arch bridge engineering.
How is linear rim velocity v related to angular velocity ω?+
Differentiating s = rθ with respect to time gives v = ds/dt = r(dθ/dt) = rω.