<p>The <strong>number 206265 arcseconds/radian is often used in astronomy for angular conversions</strong>. It is simply derived from the product of 3600 arcseconds/degree and 57.2958 degrees/radian.</p>
<h3>Edit based on symbols as defined in comments below</h3>
<p>With the distance to an object, <span class="math-container">$d$</span>, and its lateral dimension, <span class="math-container">$D$</span>, and using the small angle approximation where <span class="math-container">$D \ll d$</span>, the angular extent of the object in radians is given by <span class="math-container">$D/d$</span>. Assuming that you have a measured angle, <span class="math-container">$x$</span>, in arc seconds, you would first need to use the conversion factor above to find the angle in radians, resulting in the following equation.</p>
<p><span class="math-container">$$\frac{x}{206265} = \frac{D}{d}$$</span></p>
<p>From that relation, knowing the measured angle and distance, you can caclulate the transverse linear dimension.</p>

CC BY-SA 4.0
2020-07-03T03:41:49.757Z
d is the distance, D is the actual size, linear size, X is the angular size
Chans
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2020-07-03T03:42:47.630Z
Could you write down the formula and prove it mathematically?
Chans
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2020-07-03T21:52:10.363Z
I edited my answer based on the symbol definitions in your comment. You should edit the original question to include these definitions, as requested in several comments to the question above.
amateurAstro
CC BY-SA 4.0
2020-07-02T22:33:46.643Z
2021-06-24T08:27:06.210Z
2021-06-24T08:27:06.210Z
amateurAstro
<p>Let <span class="math-container">$d$</span> be the distance, <span class="math-container">$D$</span> is the linear size, and <span class="math-container">$x$</span> the angular size.</p>
<p>In the formula <span class="math-container">$$D = \frac{dx}{206 265}$$</span></p>
<p>what does this number mean?</p>

CC BY-SA 4.0
2020-07-03T07:54:00.933Z
@uhoh In general we want to encourage new users to post a reasonable amount of context, not just to help those who answer questions, but to help those who can learn from questions and answers (so they do not get confused with other things that have e.g. coincidental expressions or numbers).  A little too much information is always preferred over not quite enough. :-)
StephenG - Help Ukraine
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2021-06-24T08:25:32.350Z
@uhoh Your comment here was removed, so maybe you could shed some light what happened here, please? Furthermore: Are you able to point me to which "small angle formula" is refered to here?
B--rian
<definition><angular-diameter>
Why do we use the value 206265 in the small angle formula?
Chans
# Why do we use the value 206265 in the small angle formula?

(Asked by: username_0 on 2020-07-02)

Let $d$ be the distance, $D$ is the linear size, and $x$ the angular size.

In the formula $$D = \frac{dx}{206 265}$$

what does this number mean?

**Tags:** definition, angular-diameter
> **Comments:**
> **username_1**: @uhoh In general we want to encourage new users to post a reasonable amount of context, not just to help those who answer questions, but to help those who can learn from questions and answers (so they do not get confused with other things that have e.g. coincidental expressions or numbers).  A little too much information is always preferred over not quite enough. :-)  
> **username_2**: @uhoh Your comment here was removed, so maybe you could shed some light what happened here, please? Furthermore: Are you able to point me to which "small angle formula" is refered to here?  

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## 1 answers

### Answer by username_3 on 2020-07-02. Score: 4

The **number 206265 arcseconds/radian is often used in astronomy for angular conversions**. It is simply derived from the product of 3600 arcseconds/degree and 57.2958 degrees/radian.

### Edit based on symbols as defined in comments below

With the distance to an object, $d$, and its lateral dimension, $D$, and using the small angle approximation where $D \ll d$, the angular extent of the object in radians is given by $D/d$. Assuming that you have a measured angle, $x$, in arc seconds, you would first need to use the conversion factor above to find the angle in radians, resulting in the following equation.

$$\frac{x}{206265} = \frac{D}{d}$$

From that relation, knowing the measured angle and distance, you can caclulate the transverse linear dimension.

> **Comments:**
> **username_0**: d is the distance, D is the actual size, linear size, X is the angular size  
> **username_0**: Could you write down the formula and prove it mathematically?  
> **username_3**: I edited my answer based on the symbol definitions in your comment. You should edit the original question to include these definitions, as requested in several comments to the question above.  

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