Calculator
Choose one method. The calculator uses only the fields needed for that method.
Formula Used
| Case | Formula | Meaning |
|---|---|---|
| General definition | Ω = A / r² |
A is projected area. r is reference distance. |
| Cone | Ω = 2π(1 − cos(θ / 2)) |
θ is the full apex angle. |
| Spherical cap | Ω = 2π(1 − cos α) |
α is the cap half angle. |
| Circular disk | Ω = 2π(1 − d / √(d² + r²)) |
d is distance. r is disk radius. |
| Rectangle | Ω = 4 atan(ab / (d√(d² + a² + b²))) |
a and b are half width and half height. |
| Vector triangle | Ω = 2 atan2(|a · (b × c)|, abc + (a · b)c + (b · c)a + (c · a)b) |
Uses three vectors from the observer. |
How to Use This Calculator
- Select the solid angle method that matches your geometry.
- Enter the required angle, size, distance, area, or vector values.
- Keep all length inputs in the same unit system.
- Press the calculate button.
- Read steradians, square degrees, sphere fraction, and percent coverage.
- Use the chart for a quick visual comparison.
- Download the result as CSV or PDF for reports.
Example Data Table
| Example | Inputs | Formula | Solid Angle |
|---|---|---|---|
| Cone | Full angle = 60° | 2π(1 − cos 30°) | 0.841787 sr |
| Circular disk | Radius = 2, distance = 5 | 2π(1 − 5 / √29) | 0.449394 sr |
| Rectangle | Width = 4, height = 3, distance = 6 | 4 atan(ab / d√(...)) | 0.307088 sr |
| Area method | Area = 10, radius = 4 | 10 / 4² | 0.625000 sr |
| Vector triangle | (1,0,0), (0,1,0), (0,0,1) | Vector atan2 formula | 1.570796 sr |
Understanding Solid Angle
A solid angle measures how large an object appears from a point. It is the three dimensional version of a plane angle. A plane angle uses radians. A solid angle uses steradians. One full sphere around a point has four pi steradians. This makes the unit useful for optics, antennas, radiation, astronomy, sensors, lighting, and field of view work.
Why It Matters
Many measurements depend on direction and spread. A lamp sends light through a cone. A detector sees only part of a sphere. A rectangular window covers a different portion of view than a circular disk. Solid angle lets these cases share one unit. It also helps compare shapes that look different but cover similar visual space.
Main Calculation Ideas
The basic definition is simple. Solid angle equals projected surface area divided by the square of radius. In real problems, direct area is not always easy. That is why this calculator includes several models. You can use a cone angle, spherical cap angle, circular disk, rectangular aperture, surface area method, or three vectors for a spherical triangle. Each method returns steradians, square degrees, fraction of a sphere, and percent of full coverage.
Working With Inputs
Choose the shape that matches your problem. Use degrees for angular inputs. Use the same length unit for radius, distance, width, and height. The output stays valid because ratios remove the unit. For vector mode, enter three points from the observer to the triangle vertices. The tool applies a robust arctangent formula, so it works for many orientations.
Reading The Result
A larger steradian value means a larger visible region. A value near zero means a narrow view. A value near twelve point five six six means almost a complete sphere. Square degrees are helpful for astronomy and camera fields. Percent of sphere is easier for reports. Always check whether your distance is measured from the observer point to the surface center, aperture plane, or reference sphere.
Good Measurement Habits
Round only after calculation. Record assumptions beside each result. When shapes are not centered or flat, use vector geometry or a numerical surface method for stronger accuracy in final engineering checks.
FAQs
1. What is a solid angle?
A solid angle measures how much three dimensional space an object covers from a chosen point. Its unit is the steradian.
2. What is the maximum solid angle?
A full sphere around a point equals 4π steradians. That is about 12.56637 steradians.
3. When should I use the cone method?
Use the cone method when the visible region forms a circular cone from the observer. Enter the full apex angle in degrees.
4. When should I use the disk method?
Use it for a circular disk viewed along its center axis. Enter the disk radius and the observer distance from the disk center.
5. What units should I use for lengths?
Use any length unit, but keep all length inputs consistent. Ratios remove the unit, so the result stays in steradians.
6. What are square degrees?
Square degrees are another way to express angular area. They are common in astronomy, camera fields, and sky coverage reports.
7. Why use vector triangle mode?
Use vector triangle mode when three rays from the observer define a triangular patch on the unit sphere.
8. Can the area method be inaccurate?
Yes. It needs projected area at a known radius. Curved, tilted, or non-centered surfaces may need a more detailed method.