Sector Area

Definition

A sector is a “pie-slice” region of a circle bounded by two radii and an arc. The sector area is the measure of the two-dimensional space enclosed by this boundary.

Why It Matters

Sector area calculations are the ‘geometry of the slice’; they are essential for understanding partial coverage models, such as radar sweeps, pizza slices, or the spread of an irrigation pivot.

Core Concepts

• Sector Area ($A$)

• Using Radians: $A = \frac{1}{2}r^2\theta$ (where $\theta$ is in radians).

• How to read: “The A equals one-half r squared times theta.”

  • Meaning / when to use: Pie-slice area—half the radius squared times the angle in radians.

• Using Degrees: $A = \frac{\theta}{360} \cdot \pi r^2$.

• How to read: “The A equals theta divided by three-sixty times pi r squared.”

  • Meaning: Fraction of full circle area ($\pi r^2$).

• Segment of a Circle A segment is the region bounded by a chord and its arc.

• Area Calculation: $A_{\text{segment}} = A_{\text{sector}} - A_{\triangle \text{OAB}}$.

• How to read: “The area of segment equals area of sector minus area of triangle O-A-B.”

  • Meaning: Subtract the triangular portion from the pie slice to get the curved cap.

• In a $90^\circ$ sector with radius $r$: $A = \frac{1}{4}\pi r^2 - \frac{1}{2}r^2$.

• How to read: “The A equals one-fourth pi r squared minus one-half r squared.”

  • Meaning: Quarter-circle minus the inscribed right isosceles triangle.

• Annulus (The Ring) An annulus is the region between two concentric circles with radii $R$ (outer) and $r$ (inner).

• Area: $A = \pi R^2 - \pi r^2 = \pi(R^2 - r^2)$.

• How to read: “The A equals pi R squared minus pi r squared equals pi times the quantity R squared minus r squared.”

  • Meaning / when to use: Big circle minus small circle—washer, ring, or pipe cross-section area.

Connected Concepts

• Arc Length
• Circle Fundamentals
• Area of a Triangle
• Radian Measure
• Degree Measure
• Angles in a Circle