What is the difference between sector area and arc length?

Sector area is the two-dimensional area enclosed by the arc and two radii, measured in square units. Arc length is the one-dimensional length along the curved edge, measured in linear units.

How do I convert between degrees and radians?

To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. For example, 90° = π/2 radians.

What is the relationship between sector area and the full circle area?

The sector area is a fraction of the full circle area based on the angle: A_sector = (θ/360°) × A_circle when using degrees, or A_sector = (θ/(2π)) × A_circle when using radians.

Sector Area Calculator

Calculate the area of a circle sector using radius and angle in degrees or radians. This calculator provides multiple formula representations and step-by-step solutions.

Last updated: 2025-10-21 — Compiled and reviewed by Calvin (Math Research, FreeCalculators.app)

Sector Area Calculator

Radius (r)

Angle (θ)

How It Works

Sector area formula (degrees): A = (θ/360°) × πr², where θ is in degrees.

Sector area formula (radians): A = (1/2) × r² × θ, where θ is in radians.

The sector area is proportional to both the square of the radius and the angle measure.

For a full circle (360° or 2π radians), the sector area equals the full circle area: A = πr².

The relationship between sector area and arc length: A = (1/2) × r × L, where L is the arc length.

Step-by-step Examples

Example 1: Using Degrees

Calculate the sector area of a circle with radius 8 units and central angle 45°.

Given: r = 8, θ = 45°
Convert to radians: θ = 45° × π/180 = π/4 radians
Apply formula: A = (θ/360°) × πr² = (45/360) × π × 8²
Calculate: A = (1/8) × π × 64 = 8π ≈ 25.13 square units

Example 2: Using Radians

Find the sector area when radius is 6 units and angle is π/3 radians.

Given: r = 6, θ = π/3 radians
Apply formula: A = (1/2) × r² × θ = (1/2) × 6² × (π/3)
Calculate: A = (1/2) × 36 × (π/3) = 6π ≈ 18.85 square units

Common Mistakes

Forgetting to convert degrees to radians when using the radians formula.
Using the wrong formula for the given angle unit (degrees vs radians).
Confusing sector area with arc length - area is in square units, arc length is in linear units.
Not checking if the angle is in the valid range (0° to 360° or 0 to 2π radians).

Use Cases

Engineering: Calculating material requirements for circular sectors.
Architecture: Determining the area of curved sections in designs.
Physics: Analyzing angular motion and sector-based calculations.
Mathematics: Solving geometry problems involving circle sectors.
Manufacturing: Estimating material costs for sector-shaped components.

Frequently Asked Questions

Common questions about sector area calculations.

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