A graphical representation of two different steradians. The sphere has radiusr, and in this case the areaA of the highlightedspherical cap isr2. The solid angleΩ equals[A/r2] sr which is1 sr in this example. The entire sphere has a solid angle of4π sr.
Thesteradian (symbol:sr) orsquare radian[1][2] is the unit ofsolid angle in theInternational System of Units (SI). It is used inthree dimensional geometry, and is analogous to theradian, which quantifiesplanar angles. A solid angle in the form of aright circular cone can be projected onto a sphere, defining aspherical cap where the cone intersects the sphere. The magnitude of the solid angle expressed in steradians is defined as the quotient of the surface area of the spherical cap and the square of the sphere's radius. This is analogous to the way a plane angle projected onto a circle defines acircular arc on the circumference, whose length is proportional to the angle. Steradians can be used to measure a solid angle of any shape. The solid angle subtended is the same as that of a cone with the same projected area. A solid angle of one steradian subtends acone aperture of approximately 1.144 radians or 65.54 degrees.
In the SI, solid angle is considered to be adimensionless quantity, the ratio of the area projected onto a surrounding sphere and the square of the sphere's radius. This is the number of square radians in the solid angle. This means that the SI steradian is the number of square radians in a solid angle equal to one square radian, which of course is the number one. It is useful to distinguish between dimensionless quantities of a differentkind, such as the radian (in the SI, a ratio of quantities of dimension length), so the symbol sr is used. For example,radiant intensity can be measured in watts per steradian (W⋅sr−1). The steradian was formerly anSI supplementary unit, but this category was abolished in 1995 and the steradian is now considered anSI derived unit.
The namesteradian is derived from theGreekστερεόςstereos 'solid' + radian.
Solid angle of countries and other entities relative to the centre of Earth.
A steradian can be defined as the solid anglesubtended at the centre of aunit sphere by a unitarea (of any shape) on its surface. For a general sphere ofradiusr, any portion of its surface with areaA =r2 subtends one steradian at its centre.[3]
A solid angle in the form of a circular cone is related to the area it cuts out of a sphere:
Because the surface areaA of a sphere is4πr2, the definition implies that a sphere subtends4π steradians (≈ 12.56637 sr) at its centre, or that a steradian subtends1/4π ≈ 0.07958 of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is4π sr.
Section of cone (1) and spherical cap (2) that subtend a solid angle of one steradian inside a sphere
The area of aspherical cap isA = 2πrh, whereh is the "height" of the cap. IfA =r2, then. From this, one can compute thecone aperture (a plane angle)2θ of the cross-section of a simplespherical cone whose solid angle equals one steradian:
givingθ ≈ 0.572 rad = 32.77° and aperture2θ ≈ 1.144 rad = 65.54°.
The solid angle of a spherical cone whose cross-section subtends the angle2θ is:
^"Steradian",McGraw-Hill Dictionary of Scientific and Technical Terms, fifth edition, Sybil P. Parker, editor in chief. McGraw-Hill, 1997.ISBN0-07-052433-5.
^Stephen M. Shafroth, James Christopher Austin,Accelerator-based Atomic Physics: Techniques and Applications, 1997,ISBN1563964848, p. 333
^R. Bracewell, Govind Swarup, "The Stanford microwave spectroheliograph antenna, a microsteradian pencil beam interferometer"IRE Transactions on Antennas and Propagation9:1:22-30 (1961)
