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A graphical representation of 1 steradian.
The sphere has radius r, and in this case the area A of the highlighted surface patch is r2. The solid angle Ω equals A sr/r2 which is 1 sr in this example. The entire sphere has a solid angle of 4πsr.
Unit information
Unit system SI derived unit
Unit of Solid angle
Symbol ㏛ 

The steradian (symbol: sr) or square radian[1][2] 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 name is derived from the Greek stereos for "solid" and the Latin radius for "ray, beam".

The steradian, like the radian, is a dimensionless unit, essentially because a solid angle is the ratio between the area subtended and the square of its distance from the vertex: both the numerator and denominator of this ratio have dimension length squared (i.e. L2/L2 = 1, dimensionless). It is useful, however, to distinguish between dimensionless quantities of a different nature, so the symbol "sr" is used to indicate a solid angle. For example, radiant intensity can be measured in watts per steradian (W·sr−1). The steradian was formerly an SI supplementary unit, but this category was abolished in 1995 and the steradian is now considered an SI derived unit.





A is the surface area of the spherical cap, 2πrh,
r is the radius of the sphere, and
sr is the unit, steradian.

Because the surface area A of a sphere is 4πr2, the definition implies that a sphere measures 4π (≈ 12.56637) steradians. By the same argument, the maximum solid angle that can be subtended at any point is 4π sr.

Other properties[edit]

Since A = r2, it corresponds to the area of a spherical cap (A = 2πrh) (wherein h stands for the "height" of the cap), and the relationship h/r = 1/2π holds. Therefore one steradian corresponds to the plane (i.e. radian) angle of the cross-section of a simple cone subtending the plane angle 2θ, with θ given by:


A steradian is also equal to the spherical area of a polygon having an angle excess of 1 radian, to 1/4π of a complete sphere, or to (180/π)2 ≈ 3282.80635 square degrees.



Analogus to radians[edit]

In two dimensions, an angle is related to the arc length it cuts out:

l is arc length,
r is the radius of the circle, and
rad is the radian.

Instead in three dimensions, the solid angle is related to the area it cuts out:

A is the surface area of the spherical cap, 2πrh,
r is the radius of the sphere, and
sr is the steradian.

So to give an example, a measurement of the width of an object seen would be given in radians. At the same time its visible area over one's visible field would be given in steradians. Just as the area of a circle is quadratically related to its diameter or radius, so is the area of the piece of a spherical surface cut out by a circular cone related to the angle across the patch. That area, or equivalently the proportion of a sphere its cone takes up, is a measure of three dimensional proportional volume, analogous to the two dimensional measure over the circle represented by an angle.

The one dimensional circular measure has the units of radians or degrees, while the two dimensional one is expressed in steradians. In higher dimensional mathematical spaces, units for analogous solid angles have not been explicitly named. When they are used, they are dealt with by analogy with the circular or spherical cases. That is, as a proportion of the relevant unit hypersphere taken up by the generalized angle, or set expressed in spherical coordinates.

SI multiples[edit]

A complete sphere subtends 4π ≈ 12.56637 steradians, so multiples larger than the decasteradian are rarely used.

Any range in excess of the whole area of a sphere would only be needed in conjunction with non-Euclidean, spherical geometry. Then it would also usually only be required in the context of nonlocal analysis. As such, solid angles over 4π are used exceedingly rarely in the literature.

See also[edit]


