Sector Area Calculator – Find Circle Sector Area Instantly

The sector area of a circle represents a part of the total area enclosed by two radii and their intercepted arc like a “slice” of a circular pie. Whether you’re working on geometry problems, designing circular components, or analyzing angular motion, finding the sector area is a fundamental calculation in both education and engineering.

Our Sector Area Calculator on Digital Calculator allows you to compute the area of any circle sector in seconds. Simply enter the radius and angle (in degrees or radians), and get the precise sector area instantly with step-by-step explanations for complete clarity.

What Is a Sector of a Circle?

A sector is a region of a circle enclosed by two radii and the corresponding arc. It’s similar to a slice of pizza or a wedge of pie — part of the circle’s total area.

The size of a sector depends on:

• The radius (r) of the circle.
• The central angle (θ) formed by the two radii.

The larger the angle, the larger the sector area.

For a full circle (360° or 2π radians), the sector area equals the entire area of the circle.

Sector Area Formula

The formula for calculating the area of a sector depends on whether the angle is measured in degrees or radians.

1. Formula (Degrees)

A = (θ / 360) × πr²

Where:

• A = area of the sector
• θ = central angle in degrees
• r = radius of the circle

2. Formula (Radians)

A = (1/2) × r² × θ

Where:

• θ = central angle in radians

Understanding the Concept

A full circle has:

• Area = πr²
• Angle = 360° (or 2π radians)

Thus, a sector represents a fraction of the circle:

• In degrees → (θ / 360)
• In radians → (θ / 2π)

That’s why the sector’s area is proportional to its angle.

How to Use the Sector Area Calculator

Using the Digital Calculator Sector Area Calculator is simple and efficient:

1. Enter the circle’s radius (r).
   Example: 10 cm
2. Enter the central angle (θ) in degrees or radians.
   Example: 60° or π/3 radians.
3. Select the unit type.
4. Click “Calculate.”
5. Instantly view:
   • The sector area
   • The arc length
   • The full step-by-step calculation

No more manual math get accurate, detailed results in seconds.

Example 1: Using Degrees

Find the area of a sector where the radius is 8 cm and the central angle is 60°.

Formula:
A = (θ / 360) × πr²
A = (60 / 360) × π × 8²
A = (1/6) × π × 64
A = 10.67π
A ≈ 33.51 cm²

Result: The sector area = 33.51 cm²

Example 2: Using Radians

Find the area of a sector where the radius is 6 m and the central angle is 2 radians.

Formula:
A = (1/2) × r² × θ
A = 0.5 × 6² × 2
A = 0.5 × 36 × 2 = 36 m²

Result: The sector area = 36 m²

Example 3: Quarter Circle (90°)

Radius = 10 cm, Angle = 90°

A = (90 / 360) × πr²
A = (1/4) × π × 10²
A = (1/4) × 314.16
A = 78.54 cm²

Result: The area of a quarter circle = 78.54 cm²

Example 4: Semi-Circle (180°)

Radius = 12 cm, Angle = 180°

A = (180 / 360) × πr²
A = (1/2) × π × 12² = 226.19 cm²

Result: The area of the semi-circle = 226.19 cm²

Formula Relationship: Arc Length and Sector Area

The arc length (s) and sector area (A) are related through the central angle:

• Arc Length:
  s = r × θ (in radians)
• Sector Area:
  A = (1/2) × r × s

Derivation:

Since s = rθ, substitute into the formula:
A = (1/2) × r × (rθ) = (1/2) × r²θ

Thus, knowing either arc length or angle allows you to find the sector area easily — and our calculator supports both methods.

Table of Common Sector Areas (r = 10 cm)

This demonstrates how the area grows directly with the central angle.

Step-by-Step Conversion Between Radians and Degrees

To switch between radians and degrees before using the formula:

Radians to Degrees:
θ° = θ (radians) × (180 / π)

Degrees to Radians:
θ (radians) = θ° × (π / 180)

Example:
Convert 120° to radians.
θ = 120 × (π / 180) = (2π / 3) radians.

Then use:
A = (1/2) × r² × θ
A = 0.5 × 10² × (2π / 3)
A ≈ 104.72 cm²

Same result as degree method.

Real-Life Applications of Sector Area

1. Engineering and Design

Used to calculate material required for circular parts such as gears, disks, and arcs.

2. Architecture

Essential for designing arches, domes, and curved structures.

3. Physics

Applied in rotational motion, wave patterns, and angular displacement.

4. Astronomy

Used to measure planetary paths and orbital segments.

5. Everyday Geometry

From fan blades to clock dials, sector area calculations appear everywhere.

Common Mistakes in Sector Area Calculations

1. Using incorrect angle units:
   Always check if your angle is in degrees or radians before applying the formula.
2. Rounding too early:
   Keep π precise until the final step for maximum accuracy.
3. Mixing up formulas:
   Use (θ / 360) × πr² for degrees and (1/2) × r² × θ for radians.
4. Using inconsistent units:
   Keep radius and resulting area in the same unit system (cm → cm², m → m²).

The Sector Area Calculator ensures you never make these errors again.

Sector Area, Arc Length, and Segment Area – The Difference

While the sector area includes the entire slice of the circle, the segment refers only to the region between the arc and chord.

Our Arc Length Calculator and Sector Area Calculator complement each other perfectly for full circular geometry analysis.

Why Use the Sector Area Calculator

• Instant Results: Get sector area in seconds.
• Supports both degrees and radians.
• Step-by-step explanations: Learn as you calculate.
• Error-free: Avoid manual calculation mistakes.
• Educational and professional use.
• Completely free on Digital Calculator.
• Mobile-friendly: Works on all devices.

Calculate now with Digital Calculator — geometry made simple.

Why Choose Digital Calculator

At Digital Calculator, we make complex geometry effortless. Each tool is built to deliver instant accuracy, educational clarity, and user-friendly functionality. Whether you’re a student learning the fundamentals or an engineer solving real-world problems, our calculators help you work smarter and faster.From geometry to trigonometry, we simplify every equation step by step