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### Course: NASA > Unit 2

Lesson 2: Measuring the solar system- A flat earth
- Arc length
- Circumference of Earth
- Occultations
- Occultation vs. transit vs. eclipse
- Size of the moon
- Angular measure 1
- Angular measure 1
- Trigonometric ratios in right triangles
- Angular Measure 2
- Angular Measure 2
- Intro to parallax
- Parallax: distance
- Parallax method
- Solar distance
- Solve similar triangles (advanced)
- Size of the sun
- Scale of solar system

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# Angular Measure 2

## 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

where, $d$ = diameter and $D$ = distance.

This gives us the equation that we see most often:

${\theta}_{arcseconds}=\left({\displaystyle \frac{d}{D}}\right)\times 206265$

**Let’s try it out in the next exercise!**

## Want to join the conversation?

- so real distance = radius / tan(half the angular measure)

and approx.distance = diameter / tan(angular measure)

do i get it right?(14 votes)- 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 )(9 votes)

- 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)(5 votes)- It is actually: angular size in degrees = (diameter/distance) * 57.3(8 votes)

- How this result 206265 came as 57.3*3600 = 206280(7 votes)
- Its actually just an approximation, the real decimal is bigger and comes out as 206265(2 votes)

- 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. :((7 votes)
- Most of the calculations are just shortcuts. It's the same concept. Just a few technicalities like using diameters or radius, etc(1 vote)

- Under new shortcut shouldn't the second formula be:

distance to planet= diameter *57.3/angular size in degrees?(6 votes) - 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?(4 votes)
- 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)" ?(2 votes)
- in this article tan(1) is converted into a fraction

so is there any math trick available to this in our head ?(1 vote) - \theta_{arcseconds}= \left ( \dfrac{d}{D}\right)\times206265(1 vote)
- How can I measure the angle ??(1 vote)