360 degrees in a Circle. But what about a Sphere?

Factual Questions
1

A random thought crossed my mind and I realized I wasn’t all too clear on the concepts…

Now, we know that a circle consists of 360 degrees. That is, a circle drawn on any given plane can be divided into 360 degrees.

But what about a sphere ?

Consider the center point of the sphere. Now, any number of planes exist that contain the center point of the sphere. Each of these planes constitute a circle, and hence consist of 360 degrees.

So, how many degrees in a sphere ?

Is that a meaningful question ? Or is the question conceptually flawed ?

Supplementary questions: How many such planes, containing the center point, can the sphere be divided into ? Does this figure tend to infinity, or is it finite due to the fact that the sphere has a boundary, or is it undefined, or something else entirely ?

Do enlighten me. I’ll be sphere all week :wink:

2

I’m not a mathematician, but I’ll guess.

You have 360° in a circle. If each of those points represents the intersection of a circle with the plane, then you will have 360 circles perpendicular to the plane, each with 360°. 360 x 360 = 129,600.

3

Sounds infinite to me.

There are 360 degrees in a square or rectangle, 180 in a triangle, etc. - these are two-dimensional shapes, and if I’m right, you’re dealing with Euclidean geometry there. Since a sphere is three-dimensional, that might be non-Euclidean territory. I’m not sure if there is a number of degrees in a sphere.

4

I think you question “how many degrees in a sphere?” is conceptually flawed. It’s like asking “how many square inches in a cube?”

Supplementary answer: There are an infinite number of planes containing the center point. (Im’ not sure what you mean about dividing the sphere into)

5

4*pi steradians, or approximately 41,253 square degrees.

See here.

6

It isn’t really meaningful to ask how many degrees in a sphere. A degree is really a measure of angles, not circles. When we say a circle has 360 degrees, what we are really saying is the angle swept out by the circumference of the circle subtends 360 degrees.

7

If you want to define it in terms of spherical co-orindates, you end up with 2 degrees and a length vector:

1 for the angle to the horizontal plane (sigma?)
1 for the angle to the vertical plane (phi?)
1 length vector for the radii of the sphere (‘r’).

(I’m a bit fuzzy on the classical greek notion symbols)

The first two describe where you are on a sphere, the length vector describes how big the sphere is.

Compare this to your circle example in radial co-ordinates, you’ll end up with an angle and length vector.

8

When working with polyhedrons (three-dimensional solids with plane faces), you can in a sense say that they have a total of 720 degrees. To explain:

Suppose you have an infinite plane of squares. At each corner four squares meet. Each square has an angle of 90 degrees at the corner. Four corners meet to make 360 degrees. Similarly 6 triangles of 60 degrees per corner and 3 hexagons of 120 per corner also add up to 360.

Now back to our squares. Suppose we want to “fold up” some squares into a three-dimensional figure. Four squares lie flat, so we have to remove one from each corner. Now we can fold the squares into a solid with 8 vertices at which three squares meet- a cube. If you take the amount by which each vertice is less than 360 degrees (in the cube’s case, 90 degrees less) and mulitply by the 8 vertices, you get 720.

You get the same 720 result for any other polyhedron. Triangles can be folded into tetrahedrons, octohedrons, or icosahedrons, which respectively give you:
4 * (360 - [360]) = 720
6 * (360 - [460]) = 720
12 * (360 - [5*60]) = 720.

Dodecahedrons (12 pentagonal faces) give you 20 * (360 - [3*108]) = 720. It also works for semi-regular polyhedrons, or in fact any three-dimensional figure with all plane faces. The number of degrees by which all the vertices fall short of 360 will always total 720. It’s the number degrees necessary to “wrap” a plane into a three-dimensional surface.

9

I would take a WAG and say 360 squared.

That would be 360 Latitude in one axis and 360 longitude in the other axis.

Is that right?

10

As (pi) says above (I refuse to type out ll those digits. Maybe I should ll him by his forst number, 3), solid angles subtended on a sphere are measured in terms of steradians. You can look at the anguloar measure as the area on a sphere of radius R, divided by R squared. ince a full sphere has a surface area of 4(pi)R^2, the full sphere subtends 4(pi) steradians. A hemisphere is 2(pi) steradians, and so on.

Steradians are really dimensionless, since it’s the ratio of two areas. aying that steradians corresponds to so many square degrees strikes me as a little odd. I’ve never seen the expression in any math or physics text.

11

Simple® explanation of steradians

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[quote]

Use of Steradians

Date: 7/26/96 at 13:41:42
From: Anonymous
Subject: Use of Steradians

Sir,

I read your answer pertaining to solid angle measurements. I would like an example or two of how to apply steradians in real life. For instance: Let’s say an antenna radiates a beamwidth whose -3db points are 30 degrees x 40 degrees. What is the steradian angle subtended and how does it relate to the square degrees?

This leads into my real question: how many square degrees are represented by a shere (4pi sr)? Intuitively, I thought it would be 360 x 360, but think that is wrong.

Thank you,
Mike

From: Doctor Tom
Subject: Re: Use of Steradians

Hi Mike,

The way I think of steradians is that they are the surface area on the unit sphere (sphere of radius 1) that the solid angle would cut out of the surface. Thus, there are 4*pi steradians in the entire surface of the sphere.

The number of “square degrees” in a sphere is not 360x360, since even around the equator 360 degrees don’t quite fit (the tops and bottoms overlap a tiny bit). Each layer as you go toward the poles overlaps more and more. Think of the lines of latitude and longitude on the earth. For evenly spaced lines, the regions near the equator are (nearly) square, but they get skinnier and skinnier as you approach the poles, right?

So let me tell you a very interesting fact. If you draw figures on the surface of a sphere using parts of great circles (a great circle is the intersection of a plane that passes through the center of the sphere with the surface of a sphere), you get what are called “spherical triangles,” “spherical quadrilaterals,” and so on.

Let’s start with triangles. A very, very tiny triangle on the surface will be almost flat (imagine a 1 centimeter triangle on the surface of the earth - it would take amazingly accurate measurements to show that it’s not really flat). If you add the angles of a tiny triangle like that, it will be nearly 180 degrees, since it’s almost flat. The tinier it is, the closer the number will be to 180 degrees. Now consider a giant triangle that goes to the north pole, and to two points on the equator 90 degrees apart. Its angles are all 90 degrees, so this spherical triangle’s angles sum to 3*90 = 270 degrees.

The amazing fact is this: the difference between the sum of the angles of a spherical triangle and 180 degrees is the number of steradians that the triangle subtends on the surface. I should have been using radian measure - the pole-to-equator triangle has 3pi/2 as the sum of its angles. This, minus pi (180 degrees) is the number of steradians: pi/2 in this case. You can check by noticing that 8 of these triangles cover the sphere, and 8 times pi/2 is the 4pi steradians!

So, to calculate the number of steradians in a “square degree,” I’d find the equations of 4 planes that mark out a region which, when looked at from the center of the sphere, is one degree on a side. Take the dot products of the normal vectors to these planes to find the cosines of the angles between them, and then work out the angles. Add the four angles, and it will be a bit more than 2pi degrees. (Remember that in a simple quadrilateral, the interior angles add to 2pi). The difference between the sum of the four angles and 2*pi is the solid angle represented by a “square degree”. Note that your “30 degree by 40 degree” antenna problem will not make a 120 square degree solid angle. You’ve got to work it out from the equations of the planes that bound the region.

I did this a long time ago and have lost the calculations, but if you work it out in detail, you can go directly to the solid angle from the plane equations without having to mess around with an arc-cosine, but the obvious, direct method using the arc-cosine will get you there.
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12

3_14159265358979323846’s answer is correct. But since the link doesn’t seem to be working on my computer, I’ll try to explain…

A sphere isn’t a flat rectangle, so the area isn’t (height x width). But if you look at an infinitesimally small section of a sphere then you can treat it as a flat surface. What you do is integrate that over the entire sphere. If T is the “latitude” and F is the “longitude”, a tiny rectangle on a spherical surface has height dT and width cos(T) dF. So the area of the entire sphere is cos(T) dF dT integrated over F = 0 to 2 pi and T = -pi to +pi (or F=0 to 360 deg and T=-90 to +90 deg). You should get the answer mentioned by 3_1415….

The resulting quantity is called a solid angle. If you’re working with very small areas (say, size of a nebula on a telescope image) then you can use square degrees, but for larger things (e.g. beam size of a flashlight) Steradian (square radians) is more common.

13

This a correct explanation of spherical coordinates but the appropriate Greek symbols are theta and psi for the angular coordinates.

Haj

14

As I said, I’m not a mathematician; so most of the discussion has gone over my head. Can someone explain, in layman’s terms, why there are not 360 squared degrees in a sphere? Here is how I picture it:

There are 360° in a circle. There is not an infinite number of degrees. A sphere can be plotted using X, Y and Z axes. So as I said before, you can have circles 360° around the vertical axis with each one intersecting the horizontal plane. So you have 180 circles, each containing 360°. (Wait… I said 360 squared. Okay, change that to 180 x 360.) So aren’t there 64,800 whole degrees in a sphere where the smallest unit is 1°?

15

OK, so let’s divide a sphere into 180 circular strips, parallel to the “equator.” According to your way of thinking, each strip is 1 x 360 degrees and therefore equal area. But there’s a problem with that - the strips near the “poles” are much smaller than the strips near the “equator.” In fact the “area” of the circular strip is 360 x cos(T) square degrees, T being the “latitude” of that strip. So the true area of a sphere is the sum of 360 x cos(T) as T goes from -90 to +90 degrees.

16

But if you lop off the top of the sphere at latitude T, isn’t that section a circle containing 360°? That is, you can think of a sphere as being a stack of circles 1° apart with a point at the top and the bottom. Each of those circles have 360°.

17

What I mean is this: The area of a sphere contains an infinite number of points. But if one degree is the smallest unit you’re dealing with, there must be a finite number of units that will fit on the surface of the sphere.

18

Not really. It’s more like a projection of a 360 degree strip.

If you take the circular strip at 70 degree latitude and cut it into 360 equal pieces, is each piece a square? No, it’s sort of a tall trapezoid. Since it’s not a square, how can it be 1x1 degrees? It’s 1 degree tall, and width is smaller than 1 degree.

19

Let me try again. Near the equator, if you take an area 1 degree in latitude and 1 degree in longitude, it’s a square. That’s how big a square degree is. Think of that piece as a square tile, and see how many of those tiles you need to cover up the entire sphere. You need 360 of them to cover the area between 0 and 1 degree latitude. But you only need about 3 tiles to cover the area between 89 and 90 degrees latitude.

20

AS scr4 correctly points out, the units made up of intersecting degree units are not constant in area with latitude. That’s why it bothers me to talk about “square degrees”, as if you’re papering a sphere with them. The “natural” unit of measure of solid angle, the Steradian, is not a radian squared. It is defined as the ratio between the area subtended on the surface of a sphere of radius R to the square of R, just as a radian is the ratio between the length of an arc divided by the radius of the circle. That’s why a sphere has 4(pi) steradians and a circle has 2(pi) radians.

But I have trouble conceiving of what a “square degree” or a “degree squared” is. It’s nearly a square at the equator, but become trapezoidal as you move up toward the poles, and triangular near the pole. (Look at a Globe) And the surface area bunded by one degree of latitude and one degree of longitude varies significantly, becoming smaller as you go to the poles.

And if you’re not defining “degrees squared”/“square degrees” in terms of surface area on a unit sphere, then I can’t imagine what it might be.