A circular segment is a region of a circle which is "cut off" from the rest of the circle by a secant or a chord. It is the area bounded by the arc and the chord connecting the arc's endpoints.
Visually, think of it as the "crust" area if you slice a round pizza not through the center (which would make a sector), but near the edge.
Key Features
- Three Calculation Modes:
1. Radius & Angle: Ideal when you know the circle size and the sector angle.
2. Radius & Height: Useful when measuring physical objects where you can measure the depth of the arc (sagitta).
3. Chord & Height: Best for fieldwork where you can only measure the straight line width (chord) and the height of the hump.
- Major & Minor Segments: Unlike simple tools, ours allows you to toggle between the "Minor" segment (less than half the circle) and the "Major" segment (the rest of the pie).
- Interactive Visualization: A dynamic canvas draws your segment, highlighting the Chord, Height (Sagitta), and Arc Length in real-time.
- Comprehensive Results: Instantly computes Area, Arc Length, Chord Length, Height, and the Central Angle.
How to Use
1. Choose Input Mode: Select the input method that matches the data you have (e.g., "Chord & Height").
2. Enter Values: Input the required dimensions (like radius, angle, or chord length).
3. Select Type: Choose "Minor" for the small slice or "Major" for the large portion.
4. View Results: The calculate updates instantly, showing the Segment Area and all related geometric properties.
Formulas
The area of a circular segment is typically found by calculating the area of the circular sector and subtracting the area of the triangular portion.
Area Formula (using Radius R and Angle θ in radians):
Area = (R² / 2) × (θ - sin(θ))
Chord Length (c):
c = 2 × R × sin(θ / 2)
Arc Length (s):
s = R × θ
Height (h):
h = R × (1 - cos(θ / 2))
Common Uses
- Architecture: Designing arched windows, doorways, or bridges.
- Engineering: Calculating the volume of liquid in a partially filled horizontal cylindrical tank.
- Woodworking: Layout out arches for furniture or carpentry.
- Surveying: Measuring plots of land with curved boundaries.
Tips
- Units: Ensure all your input lengths are in the same unit (e.g., all inches or all meters). The calculated area will be in square units of that same type.
- Radians vs Degrees: Our calculator accepts degrees for ease of use but converts them to radians internally for calculations. If calculating manually, always double-check your angle mode.
Frequently Asked Questions (FAQs)
What is the "Height" in a circular segment?
The height is also known as the "Sagitta". It is the perpendicular distance from the midpoint of the chord to the arc of the circle.
Can I calculate the radius if I only have Chord and Height?
Yes! Our "Chord & Height" mode automatically reverse-engineers the Radius using the formula: R = (c² + 4h²) / 8h.
Conclusion
The Circular Segment Calculator bridges the gap between theoretical geometry and practical application. Whether you are building an archway or solving a math problem, it provides quick, accurate, and visual answers for all segment-related queries.
