Steradian

Solid Angle, 1 Steradian

BlankMap-World6 steradian

Steradian cone and cap

Steradian (sr) is the SI unit of solid angle. It is used in three-dimensional geometry, and is analogous to the radian which quantifies planar angles. The steradian, like the radian, is a dimensionless unit, essentially because the definition involves dividing by an area or length squared, leaving a dimensionless quantity.

Definition

A steradian is defined as the solid angle subtended at the center of a sphere by an area on the surface of the sphere that is equal to the square of the sphere's radius. In simpler terms, if you have a sphere with a radius r, and you draw a patch on its surface with an area of r^2, the solid angle facing that patch from the sphere's center is one steradian. The total solid angle around a point is 4π steradians.

Formula

The formula to calculate the solid angle in steradians is:

\[\Omega = \frac{A}{r^2}\]

where:

• \(\Omega\) is the solid angle in steradians,
• \(A\) is the surface area of the spherical cap that the solid angle subtends,
• \(r\) is the radius of the sphere.

Relation to Other Units

The steradian is related to other units of measurement in geometry and physics. For example, the total solid angle of a sphere in steradians is 4π, which connects the steradian to the π constant, a fundamental constant in mathematics. This relationship is crucial in fields such as electromagnetism, quantum physics, and astronomy, where the concepts of angles and solid angles are frequently applied.

Applications

Steradians are used in various scientific and engineering fields, including photometry, where they help quantify light emission in three dimensions, and in radiometry, where they are used to measure radiant flux. In astronomy, steradians help astronomers calculate the luminosity of stars and other celestial bodies by relating it to the solid angle they subtend as seen from Earth.

See Also

• Radian
• Sphere
• Solid angle
• Geometry
• Photometry
• Radiometry