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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.

Image Credit: Brit CruiseIt’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?

The size of the coin depends on its distance from your eye. Image Credit: Brit CruisePerhaps 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.

Angular measure

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:

Image Credit: Brit CruiseWhat 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:

Image Credit: NASA

Therefore one degree is equal to 60 x 60 = 3,600 arcseconds

Triangulation

When using angular measure we define an isosceles triangle between the observer and the sides of the object we are measuring. As follows:

Image Credit: Peter CollingridgeNotice 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?

Image Credit: Brit Cruis

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.