Sector Area Formula
Reference for sector area A = (θ/360) × πr² in degrees and A = ½r²θ in radians.
Covers arc length L = rθ with examples for pie charts and sprinkler coverage.
The Formula
A = ½r²θ (radians) or A = (θ/360) × πr² (degrees)
A sector is a "pie slice" of a circle. Its area is a fraction of the total circle area, proportional to the central angle.
Variables
| Symbol | Meaning |
|---|---|
| A | Area of the sector |
| r | Radius of the circle |
| θ | Central angle (in radians or degrees) |
Example 1
Find the area of a 90° sector with radius 6 cm
A = (90/360) × π × 6²
A = 0.25 × π × 36
A ≈ 28.27 cm²
Example 2
A sprinkler covers a sector of π/3 radians with reach of 10 m
A = ½ × 10² × (π/3)
A = ½ × 100 × 1.047
A ≈ 52.36 m²
When to Use It
Use the sector area formula when:
- Calculating the area covered by a sprinkler or radar sweep
- Designing pie charts and proportional diagrams
- Finding land area for curved property boundaries
- Computing material needed for fan-shaped or wedge-shaped pieces
Key Notes
- A full circle is a sector with θ = 2π radians — the formula gives A = ½r²(2π) = πr², confirming it matches the standard circle area formula
- A semicircle (θ = π) gives A = ½r²π = πr²/2
- If you already know the arc length s, the sector area simplifies to A = ½rs — no need to involve π separately
Key Notes
- Formula: A = ½r²θ (θ in radians): Equivalent to A = (θ / 2π) × πr² — the sector is a fraction of the full circle equal to the central angle divided by 2π. In degrees: A = (θ° / 360°) × πr².
- Connection to arc length: Arc length s = rθ and sector area A = ½r²θ = ½rs share the same central angle θ. They are both proportional to θ — doubling the angle doubles both the arc length and the area.
- Radian definition: The formula A = ½r²θ is cleanest in radians. The radian is defined precisely so that arc length = radius × angle in radians. Using degrees requires the conversion factor π/180 and makes formulas messier.
- Circular segment area: A segment is the region between a chord and its arc (sector minus the triangle formed by the two radii and the chord). A_segment = ½r²(θ − sinθ). This appears in cross-section calculations for partially filled circular pipes.
- Applications: Sector area formulas are used in engineering (belt wrap angle, gear tooth design), architecture (curved facades, domed ceilings), and data visualization (pie chart segment areas, which must be proportional to data values).