What this tool does
A circular segment is the region bounded by a chord and the arc it subtends on a circle. This calculator determines the area of that segment using geometric formulas based on the circle's radius and the central angle, or alternatively using the chord length and segment height.
The segment area differs from a sector (which includes the triangular portion to the center) by excluding the triangle formed between the chord endpoints and the circle's center. This distinction is important in engineering, architecture, and manufacturing where precise curved area calculations are required.
Common applications include calculating material needed for arched windows, determining the cross-sectional area of partially filled pipes, and computing the surface area of curved structural elements.
How it calculates
The area of a circular segment is calculated using the formula:
**A = r² × (θ - sin(θ)) / 2**
Where: - **A** = area of the segment - **r** = radius of the circle - **θ** = central angle in radians
For calculations using chord length (c) and height (h):
**r = (c²/8h) + (h/2)**
Then: **θ = 2 × arcsin(c / 2r)**
The segment area equals the sector area minus the triangular area: - Sector area = (θ/2π) × πr² = θr²/2 - Triangle area = r² × sin(θ)/2 - Segment area = r²(θ - sin(θ))/2
Who should use this
- **Architects** designing arched windows, doorways, or curved structural elements - **Civil engineers** calculating flow capacity in partially filled circular pipes and culverts - **Manufacturing engineers** determining material requirements for curved metal or glass panels - **Surveyors** computing land areas with circular boundaries - **Students** studying geometry, trigonometry, or calculus applications
Worked examples
**Example 1: Window arch calculation** A semicircular window has radius 0.5 meters. Calculate the glass area for a segment with central angle 90 degrees.
- θ = 90° = π/2 radians = 1.571 rad - A = (0.5)² × (1.571 - sin(1.571)) / 2 - A = 0.25 × (1.571 - 1) / 2 - A = 0.25 × 0.571 / 2 = **0.071 m²**
**Example 2: Pipe flow cross-section** A pipe with 300mm diameter is filled to a depth of 100mm. Find the water cross-sectional area.
- r = 150mm, h = 100mm - c = 2 × √(h × (2r - h)) = 2 × √(100 × 200) = 282.8mm - θ = 2 × arcsin(282.8 / 300) = 2.42 rad - A = 150² × (2.42 - sin(2.42)) / 2 = **18,562 mm²**
**Example 3: Pizza slice segment** A 12-inch pizza has a segment cut with chord length 8 inches and height 2 inches.
- r = (64/16) + 1 = 5 inches - θ = 2 × arcsin(4/5) = 1.85 rad - A = 25 × (1.85 - 0.96) / 2 = **11.1 square inches**
Limitations
- Central angle must be between 0° and 360° (0 and 2π radians) - Segment height cannot exceed the circle's diameter - Chord length cannot exceed the circle's diameter - Very small angles (less than 1°) may produce rounding errors in floating-point calculations - This calculator assumes a perfect circular arc; real-world curves may deviate slightly - For elliptical segments, different formulas are required
FAQs
**Q: What is the difference between a segment and a sector?** A: A sector includes the full pie slice from the center to the arc, bounded by two radii. A segment is only the region between the chord and the arc, excluding the central triangle. Segment area equals sector area minus triangle area.
**Q: How do I convert degrees to radians?** A: Multiply degrees by π/180. For example, 90° = 90 × π/180 = π/2, which is approximately 1.571 radians.
**Q: Can I calculate a segment larger than a semicircle?** A: Yes, segments can have central angles from 0° to 360°. A semicircle has θ = 180° (π radians). For angles greater than 180°, the segment is called the major segment.
**Q: Why does my segment area seem too large?** A: Ensure your angle is in the correct units. The formula requires radians, not degrees. Also verify the height does not exceed the radius.
Explore Similar Tools
Explore more tools like this one:
- Surface Area Calculator — Calculate surface area of 3D shapes with visual... - Trapezoid Area Calculator — Calculate the area of any trapezoid from its parallel... - Area Calculator - Find Area — Calculate the area of any shape including rectangles,... - 1031 Exchange Calculator — Calculate tax-deferred property exchange requirements... - 529 vs UTMA Comparison Calculator — Compare 529 plans and UTMA accounts to determine the...