🍰 Segment Area Calculator
Calculate the area of a circle segment
How to Use This Calculator
Enter Radius and Angle
Input the radius of the circle and the central angle (in degrees) that subtends the segment. Angle should be between 0 and 360 degrees.
Click Calculate
Press the "Calculate Segment Area" button to compute the area of the segment.
Review Results
View the calculated segment area, along with sector area and triangle area for reference.
Formula
Segment Area = (r²/2)(θ - sin(θ))
Where θ is in radians (convert degrees: θ_rad = θ_deg × π/180)
Alternative Form:
Segment Area = Sector Area - Triangle Area
= (θr²/2) - (r²sin(θ)/2)
= (r²/2)(θ - sin(θ))
Where:
- r = Radius of the circle
- θ = Central angle (in radians)
- sin(θ) = Sine of the angle
Example: Calculate segment area for radius = 10, angle = 60°
θ = 60° × π/180 = π/3 ≈ 1.047 radians
Segment Area = (10²/2)(1.047 - sin(1.047))
Segment Area = 50(1.047 - 0.866) = 50 × 0.181 = 9.05 square units
About Segment Area Calculator
The Segment Area Calculator helps you find the area of a circle segment - the region between a chord and its corresponding arc. A segment is formed by drawing a chord across a circle, creating two segments (major and minor).
When to Use This Calculator
- Geometry: Calculate segment areas in circle geometry problems
- Engineering: Design circular components with segments
- Architecture: Calculate areas for arched windows, doorways, or decorative elements
- Landscaping: Plan segment-shaped garden beds or pathways
- Education: Learn and practice circle segment calculations
- Construction: Calculate material requirements for segment-shaped areas
Why Use Our Calculator?
- ✅ Instant Results: Get segment area immediately
- ✅ Multiple Values: See sector area, triangle area, and segment area
- ✅ Step-by-Step Display: View the calculation formula with your values
- ✅ Works with Any Units: Meters, feet, inches, or any unit
- ✅ 100% Accurate: Precise geometric calculations
- ✅ Completely Free: No registration required
Understanding Circle Segments
A circle segment is the region between a chord and its arc. Key concepts:
- Chord: A line segment connecting two points on the circle
- Arc: The curved portion of the circle between the chord endpoints
- Segment: The area between the chord and arc
- Major vs Minor: The larger segment is the "major segment"; the smaller is the "minor segment"
- Relationship: Segment Area = Sector Area - Triangle Area
- Central Angle: The angle at the center subtended by the chord
Real-World Applications
Architecture: Calculate the area of an arched window. For a window with radius 2 meters and central angle 120°, the segment area (the glass area) is (2²/2)(2π/3 - sin(2π/3)) ≈ 2.46 square meters.
Design: When designing logo elements or graphics that use circle segments, calculate the area to determine proportions and visual balance.
Construction: Estimate material (paint, tiles, etc.) needed for segment-shaped areas in buildings or structures.
Frequently Asked Questions
What is a circle segment?
A circle segment is the region bounded by a chord and the arc between the chord's endpoints. It's essentially a sector minus the triangle formed by the chord and two radii.
What's the difference between a segment and a sector?
A sector includes the triangle from center to chord endpoints, while a segment excludes this triangle. Segment Area = Sector Area - Triangle Area.
How do I find segment area if I know chord length instead of angle?
If you know chord length (c) and radius (r), first find the angle: θ = 2 × arcsin(c/(2r)). Then use the segment area formula with this angle.
What is a major segment vs minor segment?
A chord divides a circle into two segments. The larger one (with angle > 180°) is the major segment. The smaller one (with angle < 180°) is the minor segment. This calculator finds the minor segment area.
Can the angle be greater than 180 degrees?
Yes, but this calculator finds the minor segment (the smaller area). For angles > 180°, you'd get the major segment by subtracting from full circle area, or use the formula with the complement angle.
Why does the formula use radians?
The mathematical formula works with radians. The calculator converts degrees to radians internally (θ_rad = θ_deg × π/180) before calculating.
What if the segment is a semicircle?
For a semicircle (angle = 180°), the segment area equals the sector area since the triangle has zero area (chord is a diameter). The formula still works: Area = (r²/2)(π - 0) = πr²/2.