steradian | |
---|---|

General information | |

Unit system | SI |

Unit of | solid angle |

Symbol | sr |

Conversions | |

1 sr in ... | ... is equal to ... |

SI base units | 1 m^{2}/m^{2} |

square degrees | 32400/π^{2} deg^{2}≈3282.8 deg ^{2} |

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. Whereas an angle in radians, projected onto a circle, gives a *length* of a circular arc on the circumference, a solid angle in steradians, projected onto a sphere, gives the *area* of a spherical cap on the surface. The name is derived from the Greek στερεός*stereos* 'solid' + radian.

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. L^{2}/L^{2} = 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* = *r*^{2} subtends one steradian at its centre.^{ [3] }

The solid angle is related to the area it cuts out of a sphere:

where

- Ω is the solid angle
- A is the surface area of the spherical cap, ,
- r is the radius of the sphere,
- h is the height of the cap, and
- sr is the unit, steradian.

Because the surface area *A* of a sphere is 4*πr*^{2}, the definition implies that a sphere subtends 4*π* steradians (≈ 12.56637 sr) at its centre, or that a steradian subtends 1/4*π* ≈ 0.07958 of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is 4*π* sr.

If *A* = *r*^{2}, it corresponds to the area of a spherical cap (*A* = 2*πrh*, where h is the "height" of the cap) and the relationship holds. Therefore, in this case, one steradian corresponds to the plane (i.e. radian) angle of the cross-section of a simple cone subtending the plane angle 2*θ*, with θ given by:

This angle corresponds to the plane aperture angle of 2*θ* ≈ 1.144 rad or 65.54°.

A steradian is also equal to the spherical area of a polygon having an angle excess of 1 radian, to of a complete sphere, or to 3282.80635 square degrees.

The solid angle of a cone whose cross-section subtends the angle 2*θ* is:

Millisteradians (msr) and microsteradians (μsr) are occasionally used to describe light and particle beams.^{ [4] }^{ [5] } Other multiples are rarely used.

In Euclidean geometry, an **angle** is the figure formed by two rays, called the *sides* of the angle, sharing a common endpoint, called the *vertex* of the angle. Angles formed by two rays are also known as * plane angles* as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called

The **radian**, denoted by the symbol **rad**, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius. The unit was formerly an SI supplementary unit and is currently a dimensionless SI derived unit, defined in the SI as 1 rad = 1 and expressed in terms of the SI base unit metre (m) as rad = m/m. Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing.

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a given point in space is specified by *three* numbers, : the *radial distance* of the *radial line***r** connecting the point to the fixed point of origin ; the *polar angle θ* of the

In mathematics, an **n-sphere** or **hypersphere** is an n-dimensional generalization of the 1-dimensional circle and 2-dimensional sphere to any non-negative integer *n*. The n-sphere is the setting for n-dimensional spherical geometry.

In optics, **Lambert's cosine law** says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle *θ* between the observer's line of sight and the surface normal; *I* = *I*_{0} cos *θ*. The law is also known as the **cosine emission law** or **Lambert's emission law**. It is named after Johann Heinrich Lambert, from his *Photometria*, published in 1760.

In geometry, a **solid angle** is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the *apex* of the solid angle, and the object is said to *subtend* its solid angle at that point.

In geometry, a **circular segment** or **disk segment** is a region of a disk which is "cut off" from the rest of the disk by a straight line. The complete line is known as a *secant*, and the section inside the disk as a *chord*.

A **central angle** is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one. The central angle is also known as the arc's angular distance. The arc length spanned by a central angle on a sphere is called *spherical distance*.

In radiometry, **radiance** is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiation, and to quantify emission of neutrinos and other particles. The SI unit of radiance is the watt per steradian per square metre. It is a *directional* quantity: the radiance of a surface depends on the direction from which it is being observed.

**Spherical trigonometry** is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of **spherical triangles**, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation.

A **cone** is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.

In geometry, a **spherical cap** or **spherical dome** is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single 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*.

In radiometry, **radiant intensity** is the radiant flux emitted, reflected, transmitted or received, per unit solid angle, and **spectral intensity** is the radiant intensity per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. These are *directional* quantities. The SI unit of radiant intensity is the watt per steradian, while that of spectral intensity in frequency is the watt per steradian per hertz and that of spectral intensity in wavelength is the watt per steradian per metre —commonly the watt per steradian per nanometre. Radiant intensity is distinct from irradiance and radiant exitance, which are often called *intensity* in branches of physics other than radiometry. In radio-frequency engineering, radiant intensity is sometimes called **radiation intensity**.

**Etendue** or **étendue** is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics. Other names for etendue include **acceptance**, **throughput**, **light grasp**, **light-gathering power**, **optical extent**, and the **AΩ product**. *Throughput* and *AΩ product* are especially used in radiometry and radiative transfer where it is related to the view factor. It is a central concept in nonimaging optics.

In classical mechanics, the **shell theorem** gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.

An **isotropic radiator** is a theoretical point source of waves which radiates the same intensity of radiation in all directions. It may be based on sound waves or electromagnetic waves, in which case it is also known as an **isotropic antenna**. It has no preferred direction of radiation, i.e., it radiates uniformly in all directions over a sphere centred on the source.

A **square degree** (**deg ^{2}**) is a non-SI unit measure of solid angle. Other denotations include

The **goat grazing problem** is either of two related problems in recreational mathematics involving a tethered goat grazing a circular area: the interior grazing problem and the exterior grazing problem. The former involves grazing the interior of a circular area, and the latter, grazing an exterior of a circular area. For the exterior problem, the constraint that the rope can not enter the circular area dictates that the grazing area forms an involute. If the goat were instead tethered to a post on the edge of a circular path of pavement that did not obstruct the goat, the interior and exterior problem would be complements of a simple circular area.

In geometry, a **spherical wedge** or **ungula** is a portion of a ball bounded by two plane semidisks and a spherical lune. The angle between the radii lying within the bounding semidisks is the dihedral α. If AB is a semidisk that forms a ball when completely revolved about the *z*-axis, revolving AB only through a given α produces a spherical wedge of the same angle α. Beman (2008) remarks that "a spherical wedge is to the sphere of which it is a part as the angle of the wedge is to a perigon." A spherical wedge of *α* = π radians (180°) is called a *hemisphere*, while a spherical wedge of *α* = 2π radians (360°) constitutes a complete ball.

In geometry, a **spherical sector**, also known as a **spherical cone**, is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed by the center of the sphere and the base of the cap. It is the three-dimensional analogue of the sector of a circle.

- ↑ Stutzman, Warren L; Thiele, Gary A (2012-05-22).
*Antenna Theory and Design*. ISBN 978-0-470-57664-9. - ↑ Woolard, Edgar (2012-12-02).
*Spherical Astronomy*. ISBN 978-0-323-14912-9. - ↑ "Steradian",
*McGraw-Hill Dictionary of Scientific and Technical Terms*, fifth edition, Sybil P. Parker, editor in chief. McGraw-Hill, 1997. ISBN 0-07-052433-5. - ↑ Stephen M. Shafroth, James Christopher Austin,
*Accelerator-based Atomic Physics: Techniques and Applications*, 1997, ISBN 1563964848, p. 333 - ↑ R. Bracewell, Govind Swarup, "The Stanford microwave spectroheliograph antenna, a microsteradian pencil beam interferometer"
*IRE Transactions on Antennas and Propagation***9**:1:22-30 (1961)

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