Calculator
Select a geometry model, enter values, then compute Ω in steradians.
Formula Used
- Circular cone (half-angle θ): Ω = 2π(1 − cos θ). This matches a spherical cap on the unit sphere.
- Circular disk on-axis: Ω = 2π(1 − d / √(d² + r²)). Here r is disk radius and d is distance to its center.
- Rectangle on-axis: Ω = 4 arctan( (ab) / (d √(d² + a² + b²)) ), where a=W/2 and b=H/2.
- Area on sphere: Ω = A / R², with spherical area A on a sphere of radius R.
Solid angle is dimensionless but expressed in steradians (sr). For a full sphere, Ω = 4π sr.
How to Use This Calculator
- Select a model that matches your geometry and alignment.
- Choose angle and length units that match your inputs.
- Enter dimensions, distances, or half-angle as required.
- Click Calculate to view Ω above the form.
- Use Download CSV or Download PDF for records.
For off-axis cases, use a more specialized view-factor approach.
Example Data Table
| Model | Inputs | Ω (sr) | Fraction of 4π |
|---|---|---|---|
| Cone | θ = 15° | 0.2141 | 0.0170 |
| Disk | r = 5 cm, d = 20 cm | 0.1900 | 0.0151 |
| Rectangle | W = 30 cm, H = 20 cm, d = 50 cm | 0.4560 | 0.0363 |
| Area on sphere | A = 150 cm², R = 120 cm | 0.0104 | 0.0008 |
Values are rounded and shown for demonstration.
A practical guide for reading Ω in steradians in real setups.
1) What a solid angle represents
A solid angle (Ω) tells how large an object or beam appears from a point, measured on an imaginary sphere centered at the observer. It is the 3D analogue of a planar angle: larger coverage on the sphere means a larger Ω.
2) Steradian and reference values
The unit is the steradian (sr). A full sphere contains 4π sr ≈ 12.566 sr, and a hemisphere contains 2π sr ≈ 6.283 sr. Many optical and detector setups operate at fractions of a steradian, and sr is treated as a named, dimensionless derived unit. The calculator also shows percent of the full sphere.
3) Conical beams and field of view
For a symmetric cone with half-angle θ, Ω = 2π(1 − cosθ). This model fits field-of-view limits, collimated beams, and acceptance cones in radiation work. Ω grows slowly at small θ and rises faster as θ approaches 90°.
4) Spherical-cap interpretation
The cone formula comes from a spherical cap. On a sphere of radius R, the cap area is A = 2πR²(1 − cosθ). Dividing by R² yields Ω, which explains why the result does not depend on the chosen sphere size.
5) Circular disk at a distance
For a circular aperture or detector of radius r viewed on-axis from distance d, the calculator uses Ω = 2π(1 − d/√(d² + r²)). Moving the disk closer increases Ω rapidly; in the near limit it approaches 2π sr (a hemisphere).
6) Rectangular windows and screens
For a rectangle of width W and height H centered on-axis at distance d, Ω depends on aspect ratio and distance through arctangent terms. Keep units consistent (cm, mm, or m all work) because length units cancel in the final steradian value.
7) Why solid angle matters in measurements
Solid angle is a key geometric factor in many “collected flux” problems. In radiometry, received power scales with Ω and projected area. In photometry, luminous intensity relates to flux per steradian. In detection, the fraction of 4π sr strongly influences efficiency.
8) Interpreting results and quick checks
Use percent-of-sphere to sanity-check outputs. Very small Ω (for example, below 0.01 sr) means a narrow acceptance or distant target. Values near 1 sr indicate wide coverage for planning margins. If the target is off-axis or partially blocked, a detailed view-factor approach may be needed.
FAQs
1) What is the difference between angle and solid angle?
A planar angle measures arc length on a circle, while a solid angle measures area on a sphere. Solid angle is a 3D measure of “spread” or “apparent size” from a point.
2) What does 1 steradian mean visually?
It is the solid angle that cuts out an area on a sphere equal to the square of the sphere’s radius. Since a full sphere is 4π sr, 1 sr is about 7.96% of the sphere.
3) Can Ω ever be greater than 4π?
No. From a single point, the maximum possible solid angle is the full sphere, 4π sr. If a calculation suggests more, the geometry or units are inconsistent.
4) Do I need to convert cm to meters?
No, not as long as you use the same unit for every length input in that method. Solid angle is dimensionless in the sense that length units cancel out.
5) Why does the cone method only ask for half-angle?
A perfect cone is defined by its half-angle. The sphere intersection depends only on θ, so distance does not enter when the cone apex is at the observation point.
6) Which method should I choose for a real detector?
Use “Disk” for a circular face viewed on-axis, “Rectangle” for a rectangular window centered on-axis, and “Cone” for acceptance defined by optics or collimation. If alignment is off-axis, results become approximate.
7) How accurate are the example values?
They are rounded demonstrations. Your computed Ω will depend on precise inputs and method choice. For engineering work, keep extra significant digits and validate with a known reference geometry.