What Is a Sector?
A sector is a "pie-slice" portion of a circle bounded by two radii and the arc between them. The size of a sector is determined by its radius and central angle. Sectors appear everywhere in daily life, from pizza slices and clock faces to pie charts and camera apertures.
Sector Area Formulas
Degrees Formula
When the central angle is given in degrees.
Radians Formula
When the central angle is given in radians.
Arc Length
The length of the curved edge of the sector.
How to Calculate Sector Area
Using Degrees
- Identify the radius (r) and central angle (θ) in degrees.
- Divide the angle by 360 to find the fraction of the full circle.
- Multiply by πr² (the full circle area).
Using Radians
- Identify the radius (r) and central angle (θ) in radians.
- Apply the formula A = (1/2) × r² × θ.
- This is derived from the degrees formula since θ(rad) = θ(deg) × π/180.
Example Calculation
Find the area of a sector with radius 10 and central angle 90 degrees:
- A = (90/360) × π × 10² = 0.25 × π × 100 = 25π ≈ 78.5398 square units
- Arc Length = (90/360) × 2π × 10 = 5π ≈ 15.708 units
Applications of Sector Area
Sector area calculations are used in engineering to design gears and rotating parts, in statistics for pie charts, in architecture for curved walls and domes, and in navigation for radar coverage areas. Understanding sectors is also essential for calculus problems involving polar coordinates.
Key Relationships
- A full circle is a sector with a 360-degree (2π radian) central angle.
- A semicircle is a sector with a 180-degree (π radian) central angle.
- A quadrant is a sector with a 90-degree (π/2 radian) central angle.
- The arc length of a sector equals the radius times the angle in radians.