Sector Area Calculator

**Sector area (radians):** A = (1/2) × r² × θ **Sector area (degrees):** A = (θ/360) × π × r² Or equivalently: A = (θ × π × r²) / 360. **Arc length (related):** L = r × θ (radians) = (θ/360) × 2πr (degrees) **Sector perimeter:** P = 2r + L = 2r + r × θ (radians) = 2r + (θ/360) × 2πr (degrees) Two radii + arc. **Worked examples:** **Sector with r = 10, angle = 90°:** Convert: 90° = π/2 rad. A = (1/2) × 100 × π/2 = 25π ≈ 78.54 sq units. Or: A = (90/360) × π × 100 = π × 25 = 25π. Same. This is quarter circle: area = πr²/4 = 100π/4 = 25π. ✓ **Sector with r = 6, angle = 60°:** 60° = π/3 rad. A = (1/2) × 36 × π/3 = 6π ≈ 18.85 sq units. Or fraction: 60°/360° = 1/6. (1/6) × 36π = 6π. Same. **Full circle check:** For θ = 360° = 2π rad: A = (1/2) × r² × 2π = πr². Matches standard circle area formula. ✓ **Common sector areas (r = 10):** | Angle | Fraction | Area | |---|---|---| | 30° | 1/12 | ~26.18 | | 45° | 1/8 | ~39.27 | | 60° | 1/6 | ~52.36 | | 90° | 1/4 | ~78.54 | | 120° | 1/3 | ~104.72 | | 180° | 1/2 | ~157.08 | | 270° | 3/4 | ~235.62 | | 360° | 1 (full) | ~314.16 | **Sector vs segment:** - **Sector**: includes the triangle to center (pie slice). - **Segment**: between chord and arc (lens shape). Segment area = sector area − triangle area. Triangle area (with apex at center, sides = radii): T = (1/2) × r² × sin(θ) Segment = (1/2) × r² × (θ - sin θ) [θ in radians] **Practical examples:** **Pizza slice:** 12-inch pizza (r = 6 in). 1/8 slice (45°): A = (45/360) × π × 36 = (1/8) × 113.1 ≈ 14.14 sq in. **Pie chart sector:** 40% slice of a chart with radius 5 cm: A = 0.40 × π × 25 = 31.42 cm² (just visualization, not numerical). **Circular garden bed (sector):** 10 m radius, 30° sector for flowers: A = (30/360) × π × 100 = (1/12) × 314.16 ≈ 26.18 m². Cost: 26.18 m² × $20/m² fertilizer = ~$524. **Conversions:** Degrees to radians: × π/180. Radians to degrees: × 180/π. Common: - 30° = π/6 - 45° = π/4 - 60° = π/3 - 90° = π/2 - 180° = π - 270° = 3π/2 - 360° = 2π **Relationship to pie charts:** For a pie chart with category value X out of total T: Angle = (X/T) × 360°. Area = (X/T) × πr² (where r is chart radius). For 25% of total: angle 90°, area = πr²/4. **Sector with given arc length:** If you know r and arc L: A = (1/2) × r × L Useful for road curve design where you know radius and length. **Sector applications:** - **Pie charts**: visualize proportions. - **Stadium seating**: fan-shaped sections. - **Sundials**: hour shadow areas. - **Compass roses**: directional sectors. - **Fan-shaped patios**: design and material calculation. - **Pizza/Pie cutting**: equal slices. **For full circle (θ = 360°):** Sector area = πr² (full circle area). Useful sanity check. **Common pi values:** π ≈ 3.14159 (full precision). π ≈ 3.14 (good for everyday). For sector area, π appears as factor — choose precision based on need. **Compass sector navigation:** For 30° sector in 1 km radius: A = (30/360) × π × 1 = 0.262 km² ≈ 26.2 hectares. Used in search and rescue operations. **Sector vs spherical cap:** - **Sector (2D)**: flat pie slice. - **Spherical cap (3D)**: bowl-shaped piece of sphere. Different formulas; sector is 2D. **Software:** - **Calculators**: simple formula. - **CAD**: built-in sector tools. - **GIS**: sector buffers for analyses. - **Design tools**: parametric sector shapes. **Programming (Python):** For radians: 0.5 * r**2 * theta_rad. For degrees: (theta_deg / 360) * math.pi * r**2. **Pitfalls:** - **Angle units**: must use radians in (1/2)r²θ formula. - **Confusing sector with segment**: sector includes triangle to center. - **For chord/segment**: use different formulas. - **Arc length confusion**: arc is just curve; sector is area. - **Full circle**: θ = 360° or 2π rad. - **Negative angle**: usually doesn't make sense.