A common mistake
A simple yet fundamental concept in astronomy is angular measure. It addresses a common error made by non-astronomers. To illustrate this error, try describing the size of the Sun from your point of view.
It’s tempting to say it’s about an inch wide, or the size of a quarter. There is a problem with this description however. Do you know why?
Perhaps my arm is much shorter than yours, and therefore my understanding of “a quarter” is different. The measurement depends on the exact distance to the quarter. In order to use this type of measurement we’d have to say it’s the size of a quarter observed x inches away.
Astronomers use a simpler method based on how many degrees you would tilt your telescope (or head) to scan across an object. This is known as angular measure. Click and drag the circle below to see how the angular measure changes:
This method leads to the conclusion that both the Sun and Moon are about a half of a degree in size. This means if we put 720 Moons side-by-side in a circle they would complete a ring around the sky! Convince yourself of this, it's very important:
What about measuring really tiny objects such as planets? Just as we do with microscopic objects, we simply increase the resolution of our measurement. We can divide one degree of arc into 60 arcminutes. We further divide each arcminute into 60 arcseconds:
Therefore one degree is equal to 60 x 60 = 3,600 arcseconds
When using angular measure we define an isosceles triangle between the observer and the sides of the object we are measuring. As follows:
Notice we can cut this triangle (and angle) in half to form a right triangle. We love right triangles because it allows us to use trigonometry!
tan (angular measure/2) = radius / distance
Quick review (trig in action)
Imagine a pole is 12 meters high and we have to tilt our head 36.8 degrees from the horizon to see the top. How far away are we from the pole?
tan (ABC) = opposite / adjacent
tan (36.8) = 12 / distance
distance = 12 / tan(36.8)
distance = 16 meters
Next let's review basic trigonometry & angular measure.