Sector Area Calculator

Calculate the area of a circular sector (pie slice) from radius and central angle. A = ½r²θ. Find the arc length, perimeter of the sector, and fraction of the full circle.

📐 Trigonometry📐 A = ½r²θ (θ in radians)🔢 Math

Radius r

Central angle

Angle unit

Please enter valid values.

Formula & Reference

Variable	Symbol	Formula	Units
Sector Area Calculator	—	A = ½r²θ (θ in radians)	square units

Step-by-Step Examples

Example 1

Quarter Circle

r=8, 90°.

θ=π/2
A=½×64×π/2=16π≈50.27
= ¼ of full circle (¼×πr²=¼×64π ✓)

✓ A = 50.27 sq units (quarter circle)

Example 2

Pizza Slice

r=12in, 45° (one-eighth of pizza).

A=½×144×π/4=18π≈56.55 in²
Arc length=12×π/4≈9.42 in

✓ A ≈ 56.55 in²

Example 3

Radian Input

r=5, θ=2 rad.

A=½×25×2=25 sq units
L=5×2=10 units

✓ A = 25 sq units

Real-World Applications

📐

Design

Logo design, pie charts, clock faces all use circular sectors.

🚗

Engineering

Pipe bends, cam profiles, gear tooth geometry.

🎮

Game Dev

Radar sweeps, fan-shaped attack zones, field-of-view calculations.

📊

Statistics

Pie chart slice areas proportional to their data percentages.

Common Mistakes to Avoid

⚠️

Formula requires radians

A = ½r²θ works only with θ in radians. For degrees: A = (θ/360)×πr².

⚠️

Sector ≠ segment

Sector is the pie slice (includes the triangle formed by two radii). Segment is the region between chord and arc (sector minus triangle).

⚠️

Perimeter includes two radii

Perimeter of sector = 2r + arc length = 2r + rθ. Students often forget to include the two straight edges.

Frequently Asked Questions

What is a sector? ▾

The region bounded by two radii of a circle and the arc between them. Like a pie slice.

What is the difference between major and minor sector? ▾

For angle θ < π (180°): minor sector (smaller). For θ > π: major sector (larger). They sum to the full circle.

How is sector area related to circle area? ▾

Sector area = (θ/2π) × πr² = fraction of full circle. A 90° sector = ¼ of full circle area.

What is a circular segment? ▾

The area between a chord and the arc. Segment area = sector area − triangle area = ½r²(θ − sin θ).

Application to pie charts? ▾

Each slice in a pie chart is a sector. If a category represents 30% of data, its sector has central angle 0.30 × 360° = 108° and area 30% of the circle.

Related Math Calculators

Formula Explorer connections

Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.