The steradian (symbol: sr) or square radian[1][2] is the unit of solid angle in the International System of Units (SI). It is used in three dimensional geometry, and is analogous to the radian, which quantifies planar angles. A solid angle in steradians, projected onto a sphere, gives the area of a spherical cap on the surface, whereas an angle in radians, projected onto a circle, gives a length of a circular arc on the circumference. The name is derived from the Greek στερεός stereos 'solid' + radian.
steradian | |
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General information | |
Unit system | SI |
Unit of | solid angle |
Symbol | sr |
Conversions | |
1 sr in ... | ... is equal to ... |
SI base units | 1 m2/m2 |
square degrees | 1802/π2 deg2 ≈ 3282.8 deg2 |
The steradian is a dimensionless unit, the quotient of the area subtended and the square of its distance from the centre. 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 kind, such as the radian (a ratio of quantities of dimension length), 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 steradian can be defined as the solid angle subtended at the centre of a unit sphere by a unit area on its surface. For a general sphere of radius r, any portion of its surface with area A = r2 subtends one steradian at its centre.[3]
The solid angle is related to the area it cuts out of a sphere:
where
Because the surface area A of a sphere is 4πr2, the definition implies that a sphere subtends 4π steradians (≈ 12.56637 sr) at its centre, or that a steradian subtends 1/4π ≈ 0.07958 of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is 4π sr.
The area of a spherical cap is A = 2πrh, where h is the "height" of the cap. If A = r2, then . From this, one can compute the plane aperture angle 2θ of the cross-section of a simple cone whose solid angle equals one steradian:
giving θ ≈ 0.572 rad or 32.77° and 2θ ≈ 1.144 rad or 65.54°.
The solid angle of a simple cone whose cross-section subtends the angle 2θ is:
A steradian is also equal to of a complete sphere (spat), to ≈ 3282.80635 square degrees, and to the spherical area of a polygon having an angle excess of 1 radian.[clarification needed]
Millisteradians (msr) and microsteradians (μsr) are occasionally used to describe light and particle beams.[4][5] Other multiples are rarely used.