Sector & Segment Area Calculator
Calculation Steps
📌 Definition: Sector & Segment Area
A sector is a portion of a circle defined by two radii and the connecting arc. Its area can be calculated using:
- Degrees: A = (θ/360) × π × r²
- Radians: A = ½ × r² × θ
A segment is the area between a chord and its corresponding arc. It is found by subtracting the area of the triangle (formed by the chord and radii) from the sector’s area.
📌 Key Formulas
For a circle with radius r and central angle θ (in radians):
- Sector Area (radians): A = ½ × r² × θ
- Sector Area (degrees): A = (θ/360) × π × r²
-
Circular Segment Area: A = (Sector Area) – (Triangle Area)
where Triangle Area = ½ × r² × sin(θ) (θ in radians)
📌 Examples of Calculations
| Shape | Formula | Example Calculation |
|---|---|---|
| Sector (r = 5, θ = 60°) | (60/360) × π × 5² | ≈ 13.09 |
| Sector (r = 4, θ = 1 rad) | ½ × 4² × 1 | 8 |
| Circular Segment (r = 5, θ = 1 rad) | (½×5²×1) – (½×5²×sin(1)) | ≈ 2.73 |
🔧 Practical Applications
1. Engineering & Design: Calculation of curved structures, arches, and mechanical parts.
2. Architecture: Design of domes, arches, and other curved architectural features.
3. Navigation & Surveying: Used in plotting courses and measuring land areas with curves.
❓ FAQs
Q1: What is a circular sector?
✅ A portion of a circle bounded by two radii and an arc.
Q2: How do I calculate a sector’s area?
✅ Use A = (θ/360) × π × r² for degrees or A = ½ × r² × θ for radians.
Q3: What defines a circular segment?
✅ It’s the area between a chord and the corresponding arc.
Q4: How is the area of a segment computed?
✅ Subtract the area of the triangle (½ × r² × sin(θ)) from the sector’s area.
Q5: How do I convert degrees to radians?
✅ Multiply the degree measure by π/180 to convert it to radians.