Circular Segment Area Calculator
Free circular segment area calculator. Compute the area of a circular segment from radius and height, radius and angle, or radius and chord. Shows formulas, diagrams explanation, and step-by-step results.
Input mode
Pick the measurement pair you already know. All length inputs must share the same unit.
Circle radius (R > 0).
Distance from the chord to the circle along the symmetry axis. Must satisfy 0 < h < 2R.
Use 0 < θ < 360° or 0 < θ < 2π radians for a proper segment.
Straight-line distance between the chord endpoints. Must satisfy 0 < c < 2R.
Segment area (A_seg)
Values assume a consistent length unit for all inputs.
How to Use This Calculator
Choose the pair of measurements you already know, keep your units consistent, and click Calculate to populate the metric panel on the right.
- Select whether you are supplying segment height, central angle, or chord length along with the radius.
- Enter the radius and the accompanying measurement. Angle inputs can be in degrees or radians.
- Press Calculate; the calculator displays the segment area, sector data, arc, chord, and percentage share.
- You can switch modes or adjust inputs freely; the results update automatically after each change.
Methodology
The calculator always works through the central angle θ in radians. From radius and segment height or chord length it solves the appropriate trigonometric equation (cosine for height, sine for chord) to recover θ, then applies
A_seg = (R² / 2) · (θ − sin θ)
Sector and circle areas, arc length, and perimeter follow from standard circle formulas so you can double-check your results.
Key geometry and notation
- R – circle radius.
- h – segment height (sagitta) measured from the chord to the circle along the symmetry axis.
- c – chord length (straight-line distance between chord endpoints).
- θ – central angle corresponding to the segment (radians for internal formulas).
- A_seg – area of the circular segment.
- A_sec – area of the corresponding circular sector.
Interpreting the calculator output
After a successful calculation you get:
- Segment area (A_seg) and sector area (A_sec).
- Full circle area πR².
- Segment height, chord length, arc length, and perimeter.
- Central angle expressed in both radians and degrees.
- Segment percentage of the full circle.
Full original guide (expanded)
What is a circular segment?
A circular segment is the region of a circle cut off by a chord. It is bounded by the chord on one side and the corresponding arc on the other. If you draw a chord across a circle and shade the cap between the chord and the circle, that shaded region is a circular segment. Many applications—from tank fluid levels to optics—start with the radius and one extra quantity (height, chord, or angle) and need the resulting area.
This tool reports chord length, arc length, sector area, and the segment’s share of the full circle so you can compare those quantities directly and use them for downstream engineering or educational tasks.
Core Math & Algebra tools
Circle & geometry tools
Professional use note
Suitable for engineering sketches, education, and quick design checks. For safety-critical work, validate against hand calculations, design codes, or domain-specific tools.
Circular Segment Area – FAQ
Can this calculator handle very small or very large segments?
Yes. The formulas are valid for any segment with 0 < h < 2R (or equivalently 0 < c < 2R and 0 < θ < 2π). Thin caps or almost-complete circles may show round-off noise, but the behavior remains correct. For safety-critical projects, perform an independent analytic or numerical check.
Why does the calculator need radians for the internal formulas?
Standard trigonometric formulas rely on the angle in radians because expressions like sin(θ) and θ appear naturally in calculus and geometry. You can enter degrees; the calculator converts them to radians internally and applies stable formulas.
What happens if my inputs don’t satisfy the geometric constraints?
If the chord exceeds 2R, if the height is negative or at least 2R, or if the angle lies outside the valid range, there is no real circular segment. The calculator detects domain violations and shows clear errors instead of misleading numbers.
Can I use the results to compute volumes in partially filled circular tanks?
Yes. A horizontal cylindrical tank’s liquid cross-section is a circular segment. Multiply the segment area by the tank length to get volume. This calculator focuses on the cross-section; combine it with a dedicated volume tool for full tank problems.