Steradian

The steradian is the SI

International System of Units

The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

 unit of solid angle

Solid angle

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

. It is used to describe two-dimensional angular spans

Linear span

In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...

 in three-dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

al space, analogous to the way in which the radian

Radian

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

 describes angles in a plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

. The name is derived from the Greek

Greek language

Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

 stereos for "solid" and the Latin radius for "ray, beam".

The steradian, like the radian, is dimensionless

Dimensionless quantity

In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...

 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. For 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 . Radiant intensity is distinct from irradiance and radiant exitance, which are often called intensity...

 can be measured in watts per steradian (W·sr⁻¹). The steradian was formerly an SI supplementary unit, but this category was abolished from the SI in 1995 and the steradian is now considered an SI derived unit

SI derived unit

The International System of Units specifies a set of seven base units from which all other units of measurement are formed, by products of the powers of base units. These other units are called SI derived units, for example, the SI derived unit of area is square metre , and of density is...

.

Definition

A steradian can be defined as the solid angle subtended at the center of a unit sphere

Unit sphere

In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point...

 by a unit area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

 on its surface. For a general sphere of radius

Radius

In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

 r, any portion of its surface with area A = r² subtends one steradian.

Since A = r², 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 = 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:


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

Because the surface area 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. 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 a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

 having an angle excess

Angle excess

Angle excess, also known as spherical 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. For instance, a plane triangle has an angle sum of 180°; an octant is a spherical triangle with...

 of 1 radian, to 1/(4π) of a complete sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

, or to (180/π)² ≈ 3282.80635 square degree

Square degree

A square degree is a non-SI unit measure of solid angle. It is denoted in various ways, including deg2, sq.deg. and ². Just as degrees are used to measure parts of a circle, square degrees are used to measure parts of a sphere. Analogous to one degree being equal to π /180 radians, a...

s.

The solid angle (in steradians) subtended by the cone above (whose cross-section subtends the radian angle 2θ) is given by:

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 is also called rectification of a curve. Historically, many methods were used for specific curves...

 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

Steradians only go up to 4π ≈ 12.56637, so the large multiples are not usable for the base unit, but could show up in such things as rate of coverage of solid angle, for example.

Multiple	Name	Symbol	May be visualized as...
101	decasteradian	dasr	Slightly more than the surface area of all water on Earth, relative to Earth Earth Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
100	steradian	sr	Area of Asia Asia Asia is the world's largest and most populous continent, located primarily in the eastern and northern hemispheres. It covers 8.7% of the Earth's total surface area and with approximately 3.879 billion people, it hosts 60% of the world's current human population... , relative to Earth
10−1	decisteradian	dsr	Area of Argentina Argentina Argentina , officially the Argentine Republic , is the second largest country in South America by land area, after Brazil. It is constituted as a federation of 23 provinces and an autonomous city, Buenos Aires...  + Peru Peru Peru , officially the Republic of Peru , is a country in western South America. It is bordered on the north by Ecuador and Colombia, on the east by Brazil, on the southeast by Bolivia, on the south by Chile, and on the west by the Pacific Ocean.... , relative to Earth
10−2	centisteradian	csr	Area of Paraguay Paraguay Paraguay , officially the Republic of Paraguay , is a landlocked country in South America. It is bordered by Argentina to the south and southwest, Brazil to the east and northeast, and Bolivia to the northwest. Paraguay lies on both banks of the Paraguay River, which runs through the center of the... , relative to Earth
10−3	millisteradian	msr	Area of Switzerland Switzerland Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition.... , relative to Earth
10−6	microsteradian	µsr	Area of Santa Monica, California Santa Monica, California Santa Monica is a beachfront city in western Los Angeles County, California, US. Situated on Santa Monica Bay, it is surrounded on three sides by the city of Los Angeles — Pacific Palisades on the northwest, Brentwood on the north, West Los Angeles on the northeast, Mar Vista on the east, and... , relative to Earth
10−9	nanosteradian	nsr	About 8 American football American football American football is a sport played between two teams of eleven with the objective of scoring points by advancing the ball into the opposing team's end zone. Known in the United States simply as football, it may also be referred to informally as gridiron football. The ball can be advanced by...  fields, relative to Earth
10−12	picosteradian	psr	Area of a small apartment, relative to Earth
10−15	femtosteradian	fsr	Area of a sheet of A5 paper, relative to Earth
10−18	attosteradian	asr	Area of a quarter-inch square, relative to Earth
10−21	zeptosteradian	zsr	Cross-sectional area of 32 gauge American wire gauge American wire gauge , also known as the Brown & Sharpe wire gauge, is a standardized wire gauge system used since 1857 predominantly in the United States and Canada for the diameters of round, solid, nonferrous, electrically conducting wire...  wire, relative to Earth
10−24	yoctosteradian	ysr	Surface area of a red blood cell Red blood cell Red blood cells are the most common type of blood cell and the vertebrate organism's principal means of delivering oxygen to the body tissues via the blood flow through the circulatory system... , relative to Earth