so real distance = radius / tan(half the angular measure) and approx.distance = diameter / tan(angular measure) do i get it right?

Yeah. That's a good insight. For really small angles tan x ~= x ( i.e. the limit as tan goes to 0 is the value of the angle ) (you can prove this using tan = sin /cos, using differentiation and taking limits, etc) Note: this works only if you use radians So for celestial angles, tan angle = angle, substituting in, ``` distance = diameter / tan (angle) ``` gives: ``` dist = dia / angle = 2 * radius / angle ``` For the more accurate formula, this gives: ``` dist = radius / tan ( 1/2 angle) = radius / ( 1/2 angle ) = 2 * radius / angle ``` So the 2 formulas are equivalent at small, celestial scale angles. This makes sense when we further think about it. Celestial objects are so fr away and so small, it is better to measure their size which corresponds to diameter (think, is it more useful to find the halve the size of a dot to get radius rather than a diameter?) also less calculations ( no halving the angle )

How this result 206265 came as 57.3*3600 = 206280

Its actually just an approximation, the real decimal is bigger and comes out as 206265

Hi.. In the section titled 'the new shortcut', I understood until the part where for angular measurement of 1 degree, we get: distance = 57.3*diameter However, I do not follow the next steps where we generalize angular measures. How do we obtain: angular size in degrees = diameter/(distance * 57.3)

It is actually: angular size in degrees = (diameter/distance) * 57.3

Main content

Angular Measure 2

Google Classroom

Skinny Triangle Simplification

There is one important way we can speed up our calculations using angular measure. It's an interesting trick all astronomers are familiar with. It applies when the angle of the object we are observing is very small (much less than 1 degree). This is always the case when dealing with celestial objects.

Skinny triangles have a side length which is almost the same as their heights. Compare this to wider triangles that have much longer sides as compared to the heights. Now here is the trick. If we are dealing with skinny triangles we can assume they are right triangles and use trigonometry to solve for distance. Let's review.

New shortcut

If some object has a very tiny angular size, theta, then

t, a, n, left parenthesis, theta, right parenthesis, approximately equals, theta, start superscript, equals, end superscript, start fraction, d, divided by, D, end fraction, space, space(radians)

where, d = diameter and D = distance.

1 radian = 57, point, 3 degrees = 57, point, 3, times, 3600, equals, 206265 arc seconds

This gives us the equation that we see most often:

theta, start subscript, a, r, c, s, e, c, o, n, d, s, end subscript, equals, left parenthesis, start fraction, d, divided by, D, end fraction, right parenthesis, times, 206265

Let’s try it out in the next exercise!

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cathddu
Posted 9 years ago. Direct link to cathddu's post “so real distance = radius...”
so real distance = radius / tan(half the angular measure)
and approx.distance = diameter / tan(angular measure)
do i get it right?
Button navigates to signup pageButton navigates to signup page
(14 votes)
Answer
- Ham
  Posted 7 years ago. Direct link to Ham's post “Yeah. That's a good insig...”
  Yeah. That's a good insight.
  For really small angles tan x ~= x ( i.e. the limit as tan goes to 0 is the value of the angle )
  (you can prove this using tan = sin /cos, using differentiation and taking limits, etc)
  Note: this works only if you use radians
  
  So for celestial angles, tan angle = angle, substituting in,
  distance = diameter / tan (angle)
  
  gives:
  dist = dia / angle = 2 * radius / angle
  
  For the more accurate formula, this gives:
  dist = radius / tan ( 1/2 angle) = radius / ( 1/2 angle ) = 2 * radius / angle
  
  So the 2 formulas are equivalent at small, celestial scale angles.
  
  This makes sense when we further think about it. Celestial objects are so fr away and so small, it is better to measure their size which corresponds to diameter (think, is it more useful to find the halve the size of a dot to get radius rather than a diameter?) also less calculations ( no halving the angle )
  Button navigates to signup page
  (9 votes)
Lily
Posted 6 years ago. Direct link to Lily's post “Hi.. In the section title...”
Hi..
In the section titled 'the new shortcut', I understood until the part where for angular measurement of 1 degree, we get:

distance = 57.3*diameter

However, I do not follow the next steps where we generalize angular measures. How do we obtain:

angular size in degrees = diameter/(distance * 57.3)
Button navigates to signup pageComment on Lily's post “Hi.. In the section title...”
(5 votes)
Answer
- Manish Tamhankar
  Posted 6 years ago. Direct link to Manish Tamhankar's post “It is actually: angular s...”
  It is actually: angular size in degrees = (diameter/distance) * 57.3
  Comment on Manish Tamhankar's post “It is actually: angular s...”
  (8 votes)
Anurag Kumar Das
Posted 5 years ago. Direct link to Anurag Kumar Das's post “How this result 206265 ca...”
How this result 206265 came as 57.3*3600 = 206280
Button navigates to signup pageButton navigates to signup page
(6 votes)
Answer
- meeth.chanchawat
  Posted a month ago. Direct link to meeth.chanchawat's post “Its actually just an appr...”
  Its actually just an approximation, the real decimal is bigger and comes out as 206265
  Button navigates to signup page
  (2 votes)
Himal Karkal
Posted 5 years ago. Direct link to Himal Karkal's post “Under new shortcut should...”
Under new shortcut shouldn't the second formula be:

distance to planet= diameter *57.3/angular size in degrees?
Button navigates to signup pageComment on Himal Karkal's post “Under new shortcut should...”
(6 votes)
Answer
Daniel Atwill
Posted 8 years ago. Direct link to Daniel Atwill's post “Am I stupid or something,...”
Am I stupid or something, or is it normal that I don't get this at all. I get the ideas behind it (like arcminutes and arcseconds), but have no clue what all the calculations are about, and how the new arc-measurements fit in with all this. :(
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(6 votes)
Answer
- Ham
  Posted 7 years ago. Direct link to Ham's post “Most of the calculations ...”
  Most of the calculations are just shortcuts. It's the same concept. Just a few technicalities like using diameters or radius, etc
  Button navigates to signup page
  (1 vote)
Daniel Smith
Posted 7 years ago. Direct link to Daniel Smith's post “In your formulas for dist...”
In your formulas for distance to planets, angular size in degrees, and angular measurements in arcseconds, could you please include parentheses, to make the formulas less ambiguous?
Button navigates to signup pageComment on Daniel Smith's post “In your formulas for dist...”
(4 votes)
Answer
- Swift Runner
  Posted 7 years ago. Direct link to Swift Runner's post “You can report at zendesk...”
  You can report at zendesk.
  https://khanacademy.zendesk.com/hc/en-us
  Button navigates to signup page
  (1 vote)
samo_groos
Posted 8 years ago. Direct link to samo_groos's post “In "New Shortcut": musn't...”
In "New Shortcut": musn't the formula instead of "distance to planet = (diameter * 57.3) / angular size degrees" be "distance to planet = diameter / (angular size degrees* 57.3)" ?
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(2 votes)
Answer
Smith
Posted 6 years ago. Direct link to Smith's post “in this article tan(1) is...”
in this article tan(1) is converted into a fraction
so is there any math trick available to this in our head ?
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(1 vote)
Answer
Αγνιβά Πολεμιστής Μαιτρα
Posted 7 years ago. Direct link to Αγνιβά Πολεμιστής Μαιτρα's post “How can I measure the ang...”
How can I measure the angle ??
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(1 vote)
Answer
Dhruv Rajput
Posted 5 years ago. Direct link to Dhruv Rajput's post “what actually angular siz...”
what actually angular size is? Is it the size of angle or something else?
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(1 vote)
Answer
- Suyash Shukla
  Posted 3 years ago. Direct link to Suyash Shukla's post “An object's angular size ...”
  An object's angular size is the angle between the lines of sight to its two opposite sides of a particular object.
  Button navigates to signup page
  (1 vote)

NASA

Course: NASA > Unit 2

Angular Measure 2

Skinny Triangle Simplification

New shortcut

theta, start subscript, a, r, c, s, e, c, o, n, d, s, end subscript, equals, left parenthesis, start fraction, d, divided by, D, end fraction, right parenthesis, times, 206265

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NASA

Course: NASA > Unit 2

Skinny Triangle Simplification

New shortcut

θarcseconds=(dD)×206265\theta_{arcseconds}= \left ( \dfrac{d}{D}\right)\times206265 θarcseconds​=(Dd​)×206265theta, start subscript, a, r, c, s, e, c, o, n, d, s, end subscript, equals, left parenthesis, start fraction, d, divided by, D, end fraction, right parenthesis, times, 206265

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theta, start subscript, a, r, c, s, e, c, o, n, d, s, end subscript, equals, left parenthesis, start fraction, d, divided by, D, end fraction, right parenthesis, times, 206265