Quick guide
How to use this calculator
- Choose the calculation mode when the page supports more than one direction.
- Enter the known measurements using the labels and angle units shown.
- Read the primary result first, then use the supporting rows to verify the convention or relationship.
Calculation method
Apply the selected angle relationship
Cone Ω=2π(1−cos α), where α is the half-angle; spherical region Ω=A/r².
The calculator validates geometric domains and reports degenerate vectors, impossible measurements, or unsupported ranges explicitly.
Worked example
Worked example
A cone with 60° half-angle subtends π steradians.
Cone Ω=2π(1−cos α), where α is the half-angle; spherical region Ω=A/r².
Supported inputs
Precision and limits
Angle conventions
Positive Cartesian rotations are counterclockwise. North-referenced azimuths run clockwise. Unoriented line and plane calculators return the smaller acute or right angle; directed vector and dihedral tools can return up to 180°.
Numerical range
Inputs may be zero or have absolute magnitude from 1e-100 through 1e100. General angle magnitudes are limited to 1,000,000,000 degrees so periodic reduction remains meaningful in double-precision arithmetic.
Precision
Calculations use finite double-precision arithmetic and normally display 12 significant digits. Results are numerical unless the page explicitly displays a fixed geometric identity.
Calculator-specific rule
A complete sphere is 4π steradians. Cone mode uses the angle from the cone axis to its surface, not the full apex angle.