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### 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,

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

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:

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/
```
Associated Topics:
College Higher-Dimensional Geometry
High School Higher-Dimensional Geometry

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