# **Quick angular measurement question about moon. Thanks!

by nukeman
Tags: angular, measurement, moon
P: 657
I am suppose to write a calculation on finding angular measurement of the moon. Does the following make sense, and is correct?

 Now let’s try to figure out the angular size of the moon. Lets say we are given data that states the moon has a diameter of 3476 kmm and is located 384,400 km from the earth. Now, lets take this equation step by step. Our Equation will ultimately read: 57.3 x do/d Now lets plug in out data points that have already been given to us: 57.3 x 3476/384,400 = 0.53 degrees. First, how did we get 57.3 ? Well, if you hold a ruler at arm’s length and measure the apparent size of the moon, it will measure around 7 mm, (depending on variables). Now, using the measuring apparatus we will determine the distance from the end of your thumb to the top of your shoulder (eye position). Lets use 760 mm as our distance. Now we divide our data points (7mm and 760mm) and get a estimated radians number or 0.0092 radians, which will convert into roughly 57.3 degrees. Now lets calculate: 57.3 x 3476/384,400 = 0.52 degrees (Rounded up from 0.518)
Mentor
P: 5,198
 Quote by nukeman Does the following make sense
Not entirely.

 Quote by nukeman , and is correct?
Your result is correct, yes. But I'm not convinced that you understand why.

 Quote by nukeman Our Equation will ultimately read: 57.3 x do/d Now lets plug in out data points that have already been given to us: 57.3 x 3476/384,400 = 0.53 degrees.
Assuming that do is supposed to be the physical diameter of the moon, and d is the distance to it, then yes this is the correct formula for the angular size, and since those are the values you plugged in, you got the right answer.

 Quote by nukeman First, how did we get 57.3 ? Well, if you hold a ruler at arm’s length and measure the apparent size of the moon, it will measure around 7 mm, (depending on variables). Now, using the measuring apparatus we will determine the distance from the end of your thumb to the top of your shoulder (eye position). Lets use 760 mm as our distance. Now we divide our data points (7mm and 760mm) and get a estimated radians number or 0.0092 radians, which will convert into roughly 57.3 degrees.
This is convoluted and some parts of it are just wrong. For example, 0.0092 radians is not 57.3 degrees. ONE radian is 57.3 degrees. THAT's why the 57.3 appears in the formula for the angular size -- as a conversion factor from radians to degrees.

Let me refresh your memory on how we measure angles. You can imagine drawing a radial line straight from the observer to one end of the object (in this case the moon). Then you can imagine drawing another radial line from the observer to the other end of the object. Going from one line to another, you sweep out a circular arc (a portion of a circle), since the two radii have the same length. Let's call the length of this circular arc 's', and the radial distance 'r.' The definition of the angle θ between the two lines is θ = s/r. When defined in this way, angles are measured in units of radians, which are dimensionless units (since the angles are defined as a ratio of two lengths). I've drawn a diagram to help illustrate what I mean. NOTE: using the symbols you used, s = do and r = d.

In the diagram, the two radial lines and the circular arc they cover make up a sort of pie-shaped wedge. Now, clearly, to cover an angle of 1 radian, the arc length s would have to be equal to the radius r. In that case, the pie-shaped wedge would be very close in shape to an equilateral triangle (since two sides have length r, and the third "side", which is curved, also has length r). Therefore, you would expect θ to be close to 60 degrees, but not quite. As it happens, θ = 57.3 degrees. So 1 radian = 57.3 degrees. THAT's where the 57.3 comes from. You don't need any of this nonsense about rulers and whatnot.

Since my argument about an equilateral triangle doesn't give an exact answer, I should point out that the way to get the exact conversion factor is as follows: when you sweep out a full circle, this corresponds to an angle of 2π radians. Hence, 2π radians = 360 degrees, or

1 radian = 360 degrees / 2π = 57.3 degrees.

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