Talk:Steradian

Syntax error in formula
Could someone please fix the first formula in section "Other properties"? All I can see in my browser is this error message:  Failed to parse(unknown function '\begin'): {\begin{aligned}\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{aligned}}  Thanx! 188.219.235.35 (talk) 16:08, 10 February 2014 (UTC)

Definition
So do you take a radian and revolve it around (projecting a circle), or is it a horizontal radian by a vertical radian (projecting a square)?


 * Neither. The easiest way to think about it, is to imagine projecting whatever range of angles you're interested in onto a sphere of unit radius, centered about the origin of the angles. The surface area of the total sphere is 4π units. If the projected angles "illuminate" the entire sphere, you have 4π sr. If they illuminate half the sphere (any half), you have 2π sr, and so on. It doesn't matter what the shape of the "illuminated" area is. I don't think there is a direct connection to radians, of the type you're trying to make. They are related in the sense that the perimeter of a unit circle is 2π units, and there are 2π radians in a full circle. Similarly, the surface area of a unit sphere is 4π units, and there are 4π steradians in "all directions".--Srleffler 03:32, 23 January 2006 (UTC)


 * Currently, the article would seem to contradict you (especially the figures)...? 65.183.135.231 (talk) 14:02, 29 May 2008 (UTC)

Dimensions
Even though we are thinking of a sphere, which has a curved surface, we are only thinking about the surface, and not the volume enclosed by it. Therefore solid angle is a 2-dimensional construct. Lasunncty 17:22, 3 April 2006 (UTC)
 * Solid angle is used in three-dimensional space in a manner analogous to the way that normal angle is used in two dimensional space. It defines an (infinite) volume within the larger (also infinite) space, in the same way that regular angle defines an (infinite) area within the (also infinite) plane in which the angle is defined. The steradian itself is, of course, neither 2-dimensional nor 3-dimensional. It is a dimensionless number, as noted in the article.--Srleffler 17:31, 3 April 2006 (UTC)


 * I would disagree. In the case of the circle, I think the area enclosed by an angle is not the same as the angle itself.  To me the angle is one dimension of that area, and the (infinite) radius is the other.  (I would also say that radians and steradians are unitless measurements, but not dimensionless.)  --Lasunncty 20:37, 3 April 2006 (UTC)
 * I changed the wording in the intro. See if you like the new version better.


 * Formally, the steradian is definitely dimensionless. I understand your point, though. It's actually sort of irrelevant to think about whether solid angle represents the surface area of some surface or the volume of space enclosed. Either way, it represents the ratio of either the area or the volume to the total area or volume possible, times a constant of 4&pi;. It is dimensionless like any other ratio of like-dimensioned quantities is. I think what the comment in the intro was trying to say was that you use steradians for problems in three-dimensional space, in the same way that you use radians for angles that are confined to a plane. The wording was not very good, though, and was probably not correct as written. I hope the new wording is better.--Srleffler 22:14, 3 April 2006 (UTC)

I disagree that angle and solid angle are dimensionless. Angle is a dimension in the same sense as length, time, mass, etc. The common argument that angle is dimensionless ("angle is a ratio of lengths") is flawed. It is not correct to say that "angle is the ratio of arclength to radius." Rather, it is "angle in radians is the ratio of arclength to radius." When we say "quantity (in this case, angle) in unit (in this case, radian)" we typically mean (barring offsets and nonlinear transformations, e.g., degC and dB) "quantity divided by unit," since as BIPM states (International Bureau of Weights and Measures, "The International System of Units (SI)," 8th ed., 2006.), "The value of a quantity is generally expressed as the product of a number and a unit." We all agree that the radian is a unit of angle, so "angle in radians" is dimensionless but "angle" has whatever dimension we call it to be. Let's call it what it is -- angle.

Likewise, the correct statement is "solid angle in steradians is the ratio of the area cut out of a sphere to the square of the radius." The steradian is a unit of solid angle, so "solid angle in steradians" is dimensionless. "Solid angle" has whatever dimension we call it to be -- squared angle. --daviesk24 19:18, 6 March 2014

AFAIK, Physics has proven that the radian is dimensionless (to the extent physics ever proves anything). Sisima70 (talk) 20:00, 14 March 2017 (UTC)

yarrr legibility
can someone fix the font for r^2? it took me 4 or 5 firefox ctrl-= zooms just to be able to read it (the arm of the r dives into the left crook of the 2). as it is it's illegible garbage.

here's one context/occurrence quoted from the page:

sphere having an area r²."


 * The problem is at your end. Try increasing the default font size of you rvrowser, or substituting a better one. At this end, it's plain HTML.
 * Urhixidur 02:30, 9 November 2006 (UTC)

SI multiples
I've nominated Template:SI multiples, which appears to be subst'ed into this article, for deletion. Join the discussion on WP:TFD. Han-Kwang (t) 19:52, 23 August 2007 (UTC)

Subtended in the wrong place
I think this is incorrect: A steradian is defined as the solid angle subtended at the center of a sphere of radius r by a portion of the surface of the sphere...

I think it should say something more like: A steradian is defined as the solid angle at the center of a sphere of radius r subtended by a portion of the surface of the sphere... Gwideman (talk) 02:19, 17 March 2010 (UTC)


 * I think it’s clearer in the original (subtended at) rather than angle at because the solid angle measures the area on the surface, which is not at the centre. Plus my maths dictionary also uses the subtended at wording. Vadmium (talk) 06:07, 8 June 2011 (UTC).

Error in equation?
I'm pretty sure the equation at the bottom of the definition section is incorrect, unless theta is the apex angle. If it's half the apex angle, apex angle values of pi and 2*pi don't work out as expected. —Preceding unsigned comment added by Martin.duke (talk • contribs) 19:09, 24 September 2010 (UTC)


 * I think originally the equation was correct if you noticed θ was redefined as the full aperture. Then someone “fixed” the equation for θ as defined by the cone-and-cap diagram (angle between the axis and the side of the cone). Just now I changed the definition of θ to be the same as in the diagram and clarified the aperture is 2θ. Hope it makes sense. Vadmium (talk) 06:07, 8 June 2011 (UTC).

Poor language use
This is a big problem with articles related to maths, physics and other technical fields. The language is complete gobbledegook and while it is no doubt technically and scientifically exact and correct, it serves readers of encyclopedias poorly. Encyclopedias are not simply and only reference books but the are purveyors of knowledge to the masses. Speaking as one integer member of this mass, I can't read it!

-Morris, UK — Preceding unsigned comment added by 92.20.206.0 (talk) 00:45, 16 August 2013 (UTC)

Related Links?
Assuming there is a practical purpose for this measurement, it might be useful to add a short list to the bottom of the page indicating where it is used, or even just a list of links to diciplines where it is useful. (I personally don't see any purpose to this measurement, so I'm curious to know why it would be of import.)Rashkavar (talk) 01:41, 8 November 2012 (UTC)


 * The one practical use I’m familiar with is to do with the radiant intensity link, briefly mentioned in the second paragraph. The brightness in the centre of the beam of a LED depends on the beam angle as well as the total amount of light. Vadmium (talk, contribs) 03:23, 8 November 2012 (UTC).

This article is awful.
Sorry, but a steradian is a solid angle, not the ratio of spherical-surface area to the sphere's radius. The "contributors" have really gotten carried away and have implanted a bunch of interesting but bogus notions, such as the "size" of Zimbabwe in sr. Good grief. And dump all the stuff about unit spheres. It's just confusing and irrelevant. It doesn't matter whether the sphere is a unit sphere or a non-unit sphere. A steradian is the same in either case. --MarkFilipak (talk) 03:21, 11 September 2014 (UTC)

-- agreed - much of this needs to be removed - and the stuff about surface area on earth is bogus - the earth is not a sphere — Preceding unsigned comment added by 82.8.228.52 (talk) 20:51, 16 November 2016 (UTC)

It does matter if the sphere is a unit sphere or not, since the ratio between the area and the radius of a sphere is afflicted by the rotation of the sphere (even if this is not measurable under normal circumstances). Sisima70 (talk) 20:00, 14 March 2017 (UTC)

Concept mixing
In my opinion in the legend of the first image should be A/r2 sr instead of A sr/r2, because you can't divide a unit of measure to a geometrical value. Doru001 (talk) 20:04, 8 December 2015 (UTC)

That is mathematics and apparently it is possible to do so (mathematics is experimental axiom exploration after all) Sisima70 (talk) 20:03, 14 March 2017 (UTC)

steradian
the solid angle substended made with the centre  of a sphere  by an area  aqual to square  of radius of same sphere  is called steradian  — Preceding unsigned comment added by 119.154.179.233 (talk) 14:24, 3 September 2016 (UTC)

The solid angle for a unit sphere with a variable spherical cap height, is given by $$omega=2pi*h*sqrt(2h-h^2)$$, $$omega=2pi*h*sin(arcos(1-h))$$, height h, is taken from the imagined cross-section of the unit sphere. $$omega=2pi*sin(theta)*(1-cos(theta))$$ angle theta, is obtained by measuring the arc length over the cap, from the imagined cone wall to the imagined center line of the cone (top of cap) .Cuberoottheo (talk) 13:46, 16 September 2016 (UTC)

Question
one steradian is seen at parallax of 65.54°, so an observer (standing at the center of the sphere) see a circle (just the same as moon in the night sky, but much larger) with the diameter of 65.54°, so the area of such circle would be: A = ½ ρd2 = 0.5×1.57×(65.54)2 = 3373.67121 square degrees, but in Definition section is said: "A steradian is also equal to the spherical area of a polygon having an angle excess of 1 radian, to 1/(4π) of a complete sphere, or to (180/π)2 ≈ 3282.80635 square degrees ..." how could this be happened? Tabascofernandez (talk) 01:44, 21 September 2017 (UTC)

Spheres (whether solid or their surfaces) are NOT fundamental to steradians
Although, in learning, it's helpful to picture steradians as 3-d solid chunks of a sphere extending from the center, or as areas on the surface of a sphere, steradians in fact have no dependence on spheres at all or any other bounded 3-d shape or 2-d surface. In a 3-d (Euclidian) space, any point has 4-pi steradians of solid angle around it, whether projected to infinity, to a sphere surface, a cube, or the surface of a surrounding potato. This should be clarified, technically and accurately, towards the end of the article, after the "teaching" parts that do mention spheres. A nice thing about a sphere is that surface area happens to correspond directly to solid angle, but they're not the same thing; a purist will not deign to mention spheres. (Similarly, ordinary plane angle doesn't really depend on 2-d circles or any other shapes, even though the circular compass rose is handy for showing plane angle.) — Preceding unsigned comment added by 108.73.1.253 (talk • contribs) 11 dec 2017 20:21 (UTC)

Allow far more than 13 steradians (yottateradians too) in Euclidean space
Separately, System International should not exclude steradian amounts including prefixed-amounts larger than the unit sphere. A large blanket may wrap many times around a basketball. A cabbage leaf could wrap more than once around the head of cabbage, subtending more than 13 steradians of solid angle. (Yottasteradians would imply a big cabbage.)  The current sentence in the article: "Any range in excess of the whole area of a sphere would only be needed in conjunction with non-Euclidean, spherical geometry." is wrong and should be corrected or removed. 108.73.1.253 (talk) 19:19, 11 December 2017 (UTC)

"Planck solid angle" listed at Redirects for discussion
An editor has asked for a discussion to address the redirect Planck solid angle. Please participate in the redirect discussion if you wish to do so. Andy Dingley (talk) 22:48, 26 February 2020 (UTC)

Generalization of solid angle in higher dimensions

 * https://math.stackexchange.com/questions/3439988/what-is-the-solid-angle-in-higher-dimensions — Preceding unsigned comment added by 208.98.202.34 (talk) 08:31, 12 December 2020 (UTC)
 * Is there actually a standard name for such "hypersolid angles"? Or even a standard name for their units? Double sharp (talk) 08:31, 19 August 2021 (UTC)

Coordinate Transform
Is there a way to explain the analogy between the radian and the steradian using coordinate transforms? For example, making a circle in a plane, measuring the angle, and then making another plane at a right angle to that plane and measuring the same angle in the plane that perpendicular/normal/orthogonal to the first plane? ScientistBuilder (talk) 19:32, 12 October 2021 (UTC)ScientistBuilderScientistBuilder (talk) 19:32, 12 October 2021 (UTC)