Sector Area Calculator

What Is a Sector Area?

A sector is a pie-slice region of a circle, bounded by two radii and the arc connecting them. The sector area measures how much of the total circle area falls within that slice — determined entirely by the circle's radius and the central angle of the sector. From irrigation system design to pie chart construction, sector area is a foundational concept in geometry and applied mathematics.

Sector Area Formulas

Two equivalent formulas calculate sector area, each suited to a different unit of angle measurement:

Formula Using Degrees

A = (θ / 360) × π r²

Here, θ is the central angle in degrees, r is the radius, and π ≈ 3.14159. Because a full circle spans 360°, the fraction θ/360 expresses what proportion of the complete circle the sector occupies. Multiplying that fraction by the full circle area formula π r² yields the sector area directly.

Formula Using Radians

A = ½ r² θ

When the central angle θ is in radians, the formula simplifies elegantly. A full circle spans 2π radians, so the proportional fraction is θ/(2π). Multiplying by π r² and canceling π yields ½ r² θ — a compact result with no need to reference π explicitly.

Understanding the Variables

• Radius (r): The distance from the circle's center to its edge. Any consistent unit of length (meters, feet, centimeters) is valid. Sector area scales with r², so doubling the radius quadruples the area.
• Central Angle (θ): The angle at the circle's center subtended by the sector. Valid ranges are 0°–360° in degrees or 0–2π in radians. A 90° angle equals π/2 ≈ 1.5708 radians.
• Angle Unit: Selecting the correct unit is essential. The degree formula divides by 360; the radian formula applies a ½ coefficient. Mixing units without conversion produces incorrect results.

Step-by-Step Worked Examples

Example 1 — Degrees

A circular patio has a radius of 5 meters. A decorative tile section covers a 120° sector. What is the tile area?

• A = (120 / 360) × π × 5²
• A = (1/3) × π × 25
• A = (1/3) × 78.540
• A ≈ 26.18 m²

Example 2 — Radians

A radar antenna sweeps π/4 radians and reaches 10 km. What area does it scan?

• A = ½ × 10² × (π/4)
• A = ½ × 100 × 0.7854
• A = 50 × 0.7854
• A ≈ 39.27 km²

Practical Applications

Sector area calculations appear across a wide range of professional and academic contexts:

• Agriculture: Center-pivot irrigation arms sweep circular sectors. A system with a 400-meter arm covering 270° irrigates approximately A = (270/360) × π × 400² ≈ 376,991 m² of cropland.
• Engineering: Gear teeth and rotating cam profiles rely on sector geometry for motion analysis and material volume calculations.
• Data Visualization: Pie chart segments are circular sectors sized proportionally to underlying data values — the sector area determines each slice's visual weight.
• Architecture: Curved staircases, amphitheater seating layouts, and arched facades all depend on accurate sector area measurements for material estimates.
• Astronomy: Telescope field-of-view coverage and satellite ground-track sweeps use sector area to quantify the sky regions observed during a pass.

Converting Between Degrees and Radians

To convert degrees to radians, multiply by π/180. For example, 90° × (π/180) = π/2 ≈ 1.5708 radians. To convert radians back to degrees, multiply by 180/π. Applying this conversion consistently ensures both formulas return the same numerical area for any given sector.

Methodology and Sources

The formulas and derivations on this page follow standard Euclidean geometry. The proportional derivation of the degree-based formula is thoroughly documented in Khan Academy's Sectors and Circle Areas lessons, a widely referenced resource for high school and college geometry. Worked examples and curriculum context are provided in the K20 Center's Arc Length and Sector Area geometry lesson (University of Oklahoma), which applies these concepts to real-world problems. Both sources confirm that sector area is a direct proportional application of the complete circle area formula π r².