We have an idea of why we use angular measure. Now we will see how to use it.

Keep in mind that parallax measurements only work for the nearest stars. They, therefore, form the bottom rung of what is called "the distance ladder". Other distance measurement techniques build on the parallax method. That is why we are going to explore this method in detail.

Let's start with the question: How do we measure the distance to something that is really far away?  Consider the following animation
 
if you stood at the place by the Right Angle and asked a friend to stand at the place by the Measured Angle (let's call it angle Theta), the distance between you and your friend would be the Baseline. Now, if your friend measures the angle subtended by the distance across the river, then you can calculate the distance across the river.

Most generally, when the angle is very small, and in astronomy we usually deal with very small angles, we can say that

θ   =    length of opposite side of triangle

                   length of adjacent side

So in this case,  θ = Distance Across / Baseline.


 But note that θ must be in radians, and the two lengths in the same distance units for the equation to be used. This is the Small Angle Approximation.

How do we use the Small Angle Approximation in astronomy? Two ways. First, if we observe an object from two different locations at the ends of a baseline, and measure the change in angle of the object in our field of view, we can get the distance to the object. This is Trigonometric Parallax.  The figure below shows two surveyors separated by a baseline they know, and measuring the change in angle of a tree across the river to get the distance to the tree.  Now, the opposite side of the triangle is the baseline between them, so:

Distance = Baseline / θ

For astronomy, replace the surveyors with telescopes and the tree with a planet or star.

Second, if we can measure the angle subtended by an object in space, and we know how far away it is, we can work out the size of the object. Now that the surveyors know the distance to the tree, one measures the angular size and uses the equation below to measure its height:



 Size = θ x Distance
Again, for astronomy, replace the surveyor with a telescope and the tree with a planet, moon or comet, for example.

Astronomical example:

The Sun subtends (covers) about 0.5° (0.0085 radians) of the sky. We know the sun is 1.5e8 km from the Earth. Plugging these numbers into the above Small Angle Approximation equation we get:

Size = 0.0085 radians x 1.5e8 km = 1.3e6 km in diameter (note that Size and Distance are both in km)
The actual size is slightly different but once again this is an approximation and a pretty good one at that.


So you see that even in trigonometric parallax we can get around the trigonometry and just use plain division.

Eclipses and Parallax

Let's consider one more example of parallax in astronomy. You may have heard that when a solar eclipse occurs the path of totality is very narrow. Why is that? Well, it's because of parallax. The picture below shows the predicted paths of future solar eclipses in North America. Those living along the path see the Moon aligned with the Sun. Those on either side of the path see it sufficiently offset from the Sun that the Moon does not cover all of the Sun. That is because of parallax. The Moon is so much closer to us than the Sun that viewing it from either side of the path introduces enough parallax that it appears to be displaced relative to the center of the Sun.