Angular Measure: Degrees, Minutes, and Seconds of Arc

Degrees, minutes, seconds

The apparent sizes of distant objects and things in the sky are measured by the angle they subtend at the eye. (It's a common error to suppose that the Sun, say, looks about as big across as a dinner plate; to some people, it seems bigger than that, but to others, it's smaller. So such attempts to describe apparent sizes in linear terms lead to misunderstandings and confusion. Angular sizes can be measured with instruments, and are a standard we can all agree on.)

If you are unfamiliar with angular measure, there are 90° in a right angle; 60 minutes of arc in one degree; and 60 seconds of arc in a minute. (We say “minutes of arc” to distinguish them from minutes of time.)

Minutes of arc are designated by a (') sign; so “ 30' ” is read “30 minutes”.

To give some familiar examples:

The width of your thumb, seen at arm's length, is about 2°.

The angular diameter of the Sun or the Moon is only about 1/4 of that, or just over 1/2 degree, which is 30 minutes of arc. (Yes, they look bigger than that near the horizon; the increased apparent angular size is an optical illusion.)

A person with normal vision can just distinguish two points separated by about 1' of arc. (That means you can forget about seconds of arc unless you're using a telescope.)

The angular height of mirages is always less than a degree. But, because of the horizon illusion mentioned above, people often suppose they are bigger than that.

Distant objects

Because the distances of miraged objects are always much greater than their sizes, the angles they subtend are always small. Small angles are readily related to the sizes and distances of the objects: the ratio of the size to the distance is the angular size measured in radians.

Radians

If you imagine a circle centered on your eye and passing through the object, the distance to the object is the radius of the circle, and the arc of that circle occupied by the object has the same length as the width of the object. The ratio of arc length (the size of the object) to the radius of the circle (the distance to the object) is the angular size of the object in radians.

For example, my thumb is about 2 cm (3/4 inch) wide; at arm's length it's about 60 cm (2 feet) from my eye; so its angular size is 2/60 = 1/30 of a radian. Fine. But how do we convert this to degrees?

Well, a whole circle is 360 degrees. But the arc length of the whole circle is the circumference of the circle, or 2 pi times the radius of the circle. That means 360 degrees is 2 pi = 6.28… radians. Or, one radian is 360° divided by 2 pi, or about 57.3° — nearly 60 degrees, in round numbers.

So my thumb, which is 1/30 of a radian at arm's length, is 1/30 of 60 degrees, or about 2°.

It's convenient to know that the angular diameter of the Sun or Moon is about 1/100 of a radian (about half a degree).

Small angles

Small angles are frequently measured in milliradians. A milliradian is the angle subtended by an object 1 foot across at a distance of 1000 feet, or 1 meter across at 1 kilometer. It's about 3 minutes of arc.

For very small angles, it's sometimes useful to know that 1 radian = 206265 seconds of arc, approximately.

Copyright © 2002, 2004 – 2006, 2010 Andrew T. Young


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