Measuring the Area of a Region of Sky

Date: 05/30/2003 at 11:31:28
From: Randy
Subject: Area of a Circle in Degrees

Dr. Math,

I enjoy reading your answers, and feel you are offering a great 
service. Keep up the good work.

I'm learning astronomy and have been trying understand the geometry of 
the sky. My question is this:  When I look through my telescope, what 
percentage of the sky am I looking at?  

I know the area in my field of view in either degrees x degrees or 
arcmin x arcmin. Is it possible to calculate the area of the visible 
sky, in either square degrees or square arcminutes, if there is such a 
thing? If yes, then a simple area of view/total area will give me my 
answer.

I can calculate the area when a radius is known, but what about when 
one is not known such as in degrees?  For example the area of a half 
of a circle with a radius of 1 is 6.283.

Date: 05/30/2003 at 12:27:20
From: Doctor Peterson
Subject: Re: Area of a Circle in Degrees

Hi, Randy.

I think this answer from the Dr. Math archives will help:

   Degrees in a Sphere? Steradians
   http://mathforum.org/library/drmath/view/55358.html 

You will see there that we can measure the "area" of a region of sky 
in steradians, which means the area of that region of a sphere with 
radius 1. It's easiest to work this out for a circular region that 
is cut out from the "sky" by a cone whose apex is at the center of 
the sphere. If the "radius" of your field of view (that is, half the 
angle of view) is theta, we can find the area of the "spherical cap" 
cut out by the cone on the sphere using the formula found in the Dr. 
Math Geometric Formulas FAQ:

 Sphere Formulas
 http://mathforum.org/dr.math/faq/formulas/faq.sphere.html#spherecap 

  S = 2 pi r h

The height h of the cap is 1-cos(theta), and r is taken as 1, so 
our "area" is

  2 pi (1 - cos(theta)) steradians

Since there are 4 pi steradians in a whole sphere (that is, the whole 
unit sphere has area 4 pi), this is

  (1 - cos(theta))/2

of the sky. For example, if you see two degrees across, making the 
"angular radius" of your field of view 1 degree, you are seeing

  (1 - cos(1 deg))/2 = 0.000076

of the sky, or about 1/13,000 of it.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/