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Angular measure 1

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

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 Cruise
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:
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 Collingridge
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?
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.

Want to join the conversation?

  • piceratops ultimate style avatar for user Rory Kidwell
    In the example of the 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? Would we have to take in to consideration the height of the observer eye's from the ground at (point B) the place observed from? The angle at his feet would not be 36.8 degrees to the top of the 12 feet high pole. So would the distance from the pole be wrong? Instead of tan (ABC) = opposite / adjacent; tan (36.8) = 12 / distance; distance = 12 / tan(36.8); distance = 16 meters. Would we have to know the height of eyes from ground? Would we use tan(AED) = opposite / adjacent; tan(36.8) = (12 - height of eyes from ground) / distance ; distance = (12 - height of eyes from ground) / tan(36.8); 1.3 meters high gives a distance of about 14.26 meters, 2 meters high gives a of about 13.36 meters.
    (15 votes)
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    • starky tree style avatar for user Sanjana
      Yes, sometimes the height of the observer is taken into consideration but in most cases the observer is considered as point mass if the distance being calculated is larger than the dimensions of the observer
      (8 votes)
  • old spice man green style avatar for user Jonathan Ziesmer
    Isn't the angle depicting arcminutes and arcseconds a little wide for one degree?
    (7 votes)
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  • duskpin ultimate style avatar for user Natalie L
    What is an arcminute and arc second? Also, what is tan in tan(angular measure/2)=radius/distance?
    (6 votes)
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  • blobby green style avatar for user veenajenny147
    I am still not clear about what angular measure is . Can you please make video to explain it since i am a beginner and i am starting from scratch.
    (7 votes)
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  • blobby green style avatar for user xjaross
    if 360 degrees = 24 hours why 1 degree = 60 arcminutes?
    if it have nothing to do with Earth rotation period why they call it arcminute?
    what lasts 60*360 minutes?
    (3 votes)
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  • blobby green style avatar for user lukasz.grzeczkowski
    Is there a website/calculator that would give me the angular size of the moon (as viewed from Earth of course) for different dates and if possible, to be more precise for different places in the world?
    (2 votes)
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  • aqualine seedling style avatar for user 1blackcable1
    I don't understand how the answer is 16 meters... no matter how many different methods or calculators I use I only get the answer 15.09 meters. Any help is much appreciated.
    (2 votes)
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  • duskpin sapling style avatar for user Archisman
    Why are both the lateral sides of the triangle (shown under the 'Triangulation' section) equal?
    (1 vote)
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  • blobby green style avatar for user Gaurav Kumar
    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!

    what is meant by the statement "half of a degree in size"?
    how does it lad to the conclusion about the 720 moons in 360'
    (1 vote)
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  • blobby green style avatar for user harshika.waghdhare11
    i couldn't understand,how a degree is like an arcsecond? and why the tan formula is here? couldn;t quite understand
    (1 vote)
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