Steradian

Steradian
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 4 $π$ sr.
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.

Definition

Section of cone (1) and spherical cap (2) that subtend a solid angle of one steradian inside a sphere

A steradian can be defined as the solid angle subtended at the center 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.^[3]

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

\Omega = \frac{A}{r^2}\,\mathrm{sr} \,

where

A

is the surface area of the spherical cap, 2

π

r

h

r

is the radius of the sphere, and

sr is the unit, steradian.

Because the surface area A of a sphere is 4 $π$ r², the definition implies that a sphere measures 4 $π$ (≈ 12.56637) steradians. By the same argument, the maximum solid angle that can be subtended at any point is 4 $π$ sr.

Other properties

Since A = r², it corresponds to the area of a spherical cap (A = 2 $π$ rh) (wherein h stands for the "height" of the cap), and the relationship h/r = 1/2 $π$ 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:

\begin{align} \theta & = \arccos \left( \frac{r-h}{r} \right)\\ & = \arccos \left( 1 - \frac{h}{r} \right)\\ & = \arccos \left( 1 - \frac{1}{2\pi} \right) \approx 0.572 \,\text{ rad,} \mbox{ or } 32.77^\circ. \end{align}

This angle corresponds to the plane aperture angle of 2θ ≈ 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 1/4 $π$ of a complete sphere, or to (¹⁸⁰⁄_$π$)² ≈ 3282.80635 square degrees.

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

\Omega = 2\pi\left(1 - \cos\theta\right)\,\mathrm{sr}

Analogue to radians

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

\theta = \frac{l}{r}\,\mathrm{rad}

where

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:

\Omega = \frac{A}{r^2}\,\mathrm{sr}

where

A is the surface area of the spherical cap, 2

π

rh,

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 degree, while the two dimensional one is expressed in steradians. In higher dimensional mathematical spaces, units for analogous solid angles haven't 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

A complete sphere subtends 4 $π$ ≈ 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 4 $π$ are used exceedingly rarely in the literature.

Solid angle of various areas relative to Earth

Multiple	Name	Symbol	May be visualized as...
10¹	decasteradian	dasr	Surface area of the Americas plus liquid water on Earth, relative to Earth (cyan on map).^[4] All constellations except those of the zodiac together subtend 0.992 dasr.
10⁰	steradian	sr	Area of Oceania plus Asia excluding Russia, relative to Earth (yellow on map).^[5] The Heavenly Waters constellation family subtends 1.16 sr.
10⁻¹	decisteradian	dsr	Area of Algeria plus Libya, relative to Earth (green on map).^[6] The constellation Lupus subtends 1.02 dsr.
10⁻²	centisteradian	csr	Area of Zimbabwe, relative to Earth (blue on map).^[7] The smallest constellation, Crux subtends 2.09 csr.
10⁻³	millisteradian	msr	Area of Switzerland, relative to Earth (red on map).^[8] The Earth, viewed from the Moon, subtends 1.2 msr.^[9]
10⁻⁶	microsteradian	µsr	Area of Costa Mesa, California, relative to Earth.^[10] The Sun and the Moon, viewed from Earth, each subtends 60 µsr.
10⁻⁹	nanosteradian	nsr	About 8 American football fields, relative to Earth. Mars, viewed from Earth at its closest approach, subtends 11 nsr.^[11]
10⁻¹²	picosteradian	psr	Area of a small apartment, relative to Earth. Pluto, viewed from Earth at its closest approach, subtends 0.24 psr.^[12]
10⁻¹⁵	femtosteradian	fsr	Area of a sheet of A5 paper, relative to Earth. Alpha Centauri A, viewed from Earth, subtends 0.9 fsr.^[13]
10⁻¹⁸	attosteradian	asr	Area of a quarter-inch square, relative to Earth. Proxima Centauri, viewed from Earth, subtends 20 asr.^[14]
10⁻²¹	zeptosteradian	zsr	Cross-sectional area of 32 gauge wire, relative to Earth.
10⁻²⁴	yoctosteradian	ysr	Surface area of a red blood cell, relative to Earth.

Notes and references

↑ 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.
↑ 10.0 sr (404 million km² out of 510 million km²)
↑ 1.01 sr (40.8 million km² out of 510 million km²)
↑ 0.102 sr (4.14 million km² out of 510 million km²)
↑ 0.00963 sr (391 000 km² out of 510 million km²); Paraguay, at 0.0100 sr (407 000 km²) is closer to 1 csr, but has been shaded for the 10 sr region as part of the Americas
↑ 0.00102 sr (41 300 km² out of 510 million km²)
↑ Near-side/far-side impact crater counts | NASA Lunar Science Institute
↑ 0.00000100 sr (40.7 km² out of 510 million km²)
↑ $π$ × (25.113 / 60 / 60 / 2)² / 3282.80635 × 1 000 000 000
↑ $π$ × (0.115 / 60 / 60 / 2)² / 3282.80635 × 1 000 000 000
↑ $π$ × (0.007 / 60 / 60 / 2)² / 3282.80635 × 1 000 000 000
↑ $π$ × (0.001 / 60 / 60 / 2)² / 3282.80635 × 1 000 000 000

SI units

Base units	ampere candela kelvin kilogram metre mole second

Derived units with special names	becquerel coulomb degree Celsius farad gray henry hertz joule katal lumen lux newton ohm pascal radian siemens sievert steradian tesla volt watt weber

Other accepted units	astronomical unit dalton day decibel degree of arc electronvolt hectare hour litre minute minute of arc neper second of arc tonne atomic units natural units

See also	Conversion of units History of the metric system Metric prefixes Proposed redefinitions Systems of measurement

Book Category Outline

Classical mechanics SI units

Dimensions	—	L	L²	Dimensions	—	—	—
Linear/translational quantities				Angular/rotational quantities
T	time: $t$ s	absement: $A$ m s		T	time: $t$ s
—		distance: $d$ , position: $r$ , $s$ , $x$ , displacement m	area: $A$ m²	—		angle: $θ$ , angular displacement: $θ$ rad	solid angle: $Ω$ rad², sr
T⁻¹	frequency: $f$ s⁻¹, Hz	speed: $v$ , velocity: $v$ m s⁻¹	kinematic viscosity: $ν$ , specific angular momentum: $h$ m² s⁻¹	T⁻¹	frequency: $f$ s⁻¹, Hz	angular speed: $ω$ , angular velocity: $ω$ rad s⁻¹
T⁻²		acceleration: $a$ m s⁻²		T⁻²		angular acceleration: $α$ rad s⁻²
T⁻³		jerk: $j$ m s⁻³		T⁻³		angular jerk: $ζ$ rad s⁻³

M	mass: $m$ kg			ML²	moment of inertia: $I$ kg m²
MT⁻¹		momentum: $p$ , impulse: $J$ kg m s⁻¹, N s	action: $𝒮$ , actergy: $ℵ$ kg m² s⁻¹, J s	ML²T⁻¹		angular momentum: $L$ , angular impulse: $Δ L$ kg m² s⁻¹	action: $𝒮$ , actergy: $ℵ$ kg m² s⁻¹, J s
MT⁻²		force: $F$ , weight: $F g$ kg m s⁻², N	energy: $E$ , work: $W$ kg m² s⁻², J	ML²T⁻²		torque: $τ$ , moment: $M$ kg m² s⁻², N m	energy: $E$ , work: $W$ kg m² s⁻², J
MT⁻³		yank: $Y$ kg m s⁻³, N s⁻¹	power: $P$ kg m² s⁻³, W	ML²T⁻³		rotatum: $P$ kg m² s⁻³, N m s⁻¹	power: $P$ kg m²s⁻³, W

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Steradian

Definition

Other properties

Analogue to radians

SI multiples

See also

Notes and references