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Steradian
 
 
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The steradian (symbol: sr) is the SI
Si

Si, si, or SI may refer to :...
 unit of solid angle
Solid angle

The solid angle, O, is the angle in three-dimensional space that an object subtends at a point. It is a measure of how big that object appears to an observer looking from that point....
. It is used to describe two-dimensional angular spans in three-dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al space, analogous to the way in which the radian
Radian

The radian is a unit of plane angle, equal to 180/pi Degree , or about 57.2958 degrees, or about 57?17'45?. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
 describes angles in a plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
. The name is derived from the Greek
Greek language

Greek is an Indo-European languages native to the southern Balkan peninsula, the language of the Greek people. It forms an independent branch within Indo-European....
 stereos for "solid" and the Latin radius for "ray, beam".

The steradian, like the radian, is dimensionless because 1 sr = m2·m-2 = 1.






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Steradian
The steradian (symbol: sr) is the SI
Si

Si, si, or SI may refer to :...
 unit of solid angle
Solid angle

The solid angle, O, is the angle in three-dimensional space that an object subtends at a point. It is a measure of how big that object appears to an observer looking from that point....
. It is used to describe two-dimensional angular spans in three-dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al space, analogous to the way in which the radian
Radian

The radian is a unit of plane angle, equal to 180/pi Degree , or about 57.2958 degrees, or about 57?17'45?. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
 describes angles in a plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
. The name is derived from the Greek
Greek language

Greek is an Indo-European languages native to the southern Balkan peninsula, the language of the Greek people. It forms an independent branch within Indo-European....
 stereos for "solid" and the Latin radius for "ray, beam".

The steradian, like the radian, is dimensionless because 1 sr = m2·m-2 = 1. It is useful, however, to distinguish between dimensionless quantities of different nature, so in practice the symbol "sr" is used where appropriate, rather than the derived unit "1" or no unit at all. As an example, radiant intensity
Radiant intensity

In radiometry, radiant intensity is a measure of the intensity of electromagnetic radiation. It is defined as power per unit solid angle. The SI unit of radiant intensity is watts per steradian ....
 can be measured in watts per steradian (W·sr-1).

Definition

A single unit of steradian is defined as the solid angle
Solid angle

The solid angle, O, is the angle in three-dimensional space that an object subtends at a point. It is a measure of how big that object appears to an observer looking from that point....
 subtended
Subtended

In geometry, an angle subtended by an arc is one whose two rays pass through the endpoints of the arc. The precise meaning varies with the context....
 at the center of a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 of radius
RADIUS

Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
 r by a portion of the surface of the sphere having an area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
 r2.

If this area, A, is equal to r2 and it corresponds to the area of a spherical cap
Spherical cap

In geometry, a spherical cap is a portion of a sphere cut off by a Plane . If the plane passes through the center of the sphere, so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere....
 (A = 2prh,) then the relationship holds. Then the solid angle of the simple cone subtending an angle ? is equal to:

This angle corresponds to an apex angle of 2? ˜ 1.144 rad or 65.54°.

Because the surface area of this sphere is 4pr2, the definition implies that a sphere measures 4p  = 12.56637 steradians. By the same argument, the maximum solid angle that can be subtended at any point is 4psr. A steradian can also be called a squared radian.

A steradian is also equal to the spherical area of a polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
 having an angle excess
Angle excess

Angle excess is the amount by which the sum of the angles of a polygon on a sphere exceeds the sum of the angles of a polygon with the same number of sides in a plane ....
 of 1 radian, to 1/(4p
Pi

Pi or p is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean geometry; this is the same value as the ratio of a circle's area to the square of its radius....
) of a complete sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
, or to (180/p)² or 3282.80635 square degree
Square degree

A square degree is a non-SI Units of measurement measure of solid angle. Just as degree s are used to measure parts of a circle, square degrees are used to measure parts of a sphere....
s.

The steradian was formerly an SI supplementary unit
SI supplementary unit

Until 1995, SI supplementary units were:As of October 1995, the category of "supplementary units" has been abolished from the SI system of measurement, and the radian and the steradian are now considered SI derived units....
, but this category was abolished from the SI
Si

Si, si, or SI may refer to :...
 in 1995 and the steradian is now considered an SI derived unit
SI derived unit

SI derived units are part of the SI system of measurement Units of measurements and are derived from the seven SI base units.Note that while the names of all SI units are in lowercase, the symbols of units named after people are written with an initial capital letter ....
.

Analogue to radians


In two dimensions, the angle in radians is related to the arc length
Arc length

Determining the length of an irregular arc segment ? also called rectification of a curve ? was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form expression in some cases....
 it cuts out:
where
l is arc length, and r is the radius of the circle.

Now in three dimensions, the solid angle in steradians is related to the area it cuts out:
where
S is the surface area, and r is the radius of the sphere.

SI multiples

Multiple Name Symbol
100 steradiansr
10–1 decisteradiandsr
10–2 centisteradian csr
10–3 millisteradian msr
10–6 microsteradian µsr
10–9 nanosteradiannsr
10–12 picosteradianpsr
10–15 femtosteradianfsr
10–18 attosteradianasr
10–21 zeptosteradian zsr
10–24 yoctosteradian ysr


See also


  • Solid angle
    Solid angle

    The solid angle, O, is the angle in three-dimensional space that an object subtends at a point. It is a measure of how big that object appears to an observer looking from that point....