<p>Using basic circular maths: where $d$ is the distance of the star from the observer in AU, and $r$ is the star's radius in AU, and $a$ is the angle encompassed by the radius of the star in degrees:</p>

<p>$
r = \frac{a}{360}2\pi{d}
$</p>

<p>Now rearrange it to make $a$ the subject:</p>

<p>$
a = \frac{180r}{\pi{d}}
$</p>

<p>To get $a$ in arcseconds, you need to multiply the result by 3600 (because there are 3600 arcseconds in a degree):</p>

<p>$
a = \frac{648000r}{\pi{d}}
$</p>

<p>Now, convert your numbers from cm and parsecs to astronomical units:</p>

<p>$r$ = 3.18e13cm = 2.126 AU</p>

<p>$d$ = 220pc = 4.538e7 AU</p>

<p>Put them into the equation:</p>

<p>$
a = \frac{648000\times2.126}{{4.538\times10^7}\pi}
$</p>

<p>To get:</p>

<p>$
a = 0.00966327′′
$</p>

<p>Hope that helps!</p>

<p><strong>Edit:</strong></p>

<p>As Mike has pointed out, the final equation I've ended with can be further simplified if you use different units, to:</p>

<p>$
a = \frac{r}{d}
$</p>

<p>where $r$ is in AU and $d$ is in parsecs, giving an answer $a$ in arcseconds.</p>

<p>This is because the equation $a = \frac{648000r}{\pi{d}}$ has $\frac{648000}{\pi}$ in it - which is the definition of a parsec in AU.  Therefore, multiplying $d$ by this number to convert it from AU to parsecs, we get $a = \frac{648000r}{648000/\pi\times\pi{d}}$. Cancel out the pi on the bottom, and cancel the ${648000}$ on the top and bottom, and you're left with $a = \frac{r}{d}$ where $r$ is in au and $d$ is in pc.</p>

CC BY-SA 3.0
2018-03-01T15:58:31.277Z
I got the same result by using $a_{as} = r_{AU} / d_{pc}$.
Mike G
CC BY-SA 3.0
2018-03-01T16:54:05.647Z
@MikeG Yes, I wasn't aware there was an actual equation for it; I just used my (limited) knowledge of how circles work to find an equation. I guess your one's more succinct! It makes sense, because the equation $a = \frac{648000r}{\pi{d}}$ has $\frac{648000}{\pi}$ in it - which is the definition of a parsec in AU.  Therefore, multiplying $d$ by this number to convert it to parsecs, we get $a = \frac{648000r}{648000/\pi\times\pi{d}}$. Cancel out the pi on the bottom, and cancel the ${648000}$ on the top and bottom, and you're left with $a = \frac{r}{d}$ where $r$ is in au and $d$ is in pc.
JThistle
CC BY-SA 3.0
2018-03-05T10:43:22.267Z
`+1` Working things though step-by-step and explaining along the way is a great way to answer, and can be very helpful for the OP or for future readers.
uhoh
CC BY-SA 3.0
2018-03-01T11:16:08.010Z
2018-03-01T16:59:04.737Z
2018-03-01T16:59:04.737Z
JThistle
<pre><code>2 * arctan(r / d)
</code></pre>

<p>Use Wolfram Alpha, it takes free-form units like cm, parsecs, etc and usually does the right thing. And you can specify the unit for the answer you're looking for (e.g. arcseconds).</p>

<p><a href="http://www.wolframalpha.com/input/?i=2+*+arctan(3.18e13+cm+%2F+220+parsecs)+in+arcseconds" rel="nofollow noreferrer">http://www.wolframalpha.com/input/?i=2+*+arctan(3.18e13+cm+%2F+220+parsecs)+in+arcseconds</a></p>

<p>In this case, the answer is 0.019 arc seconds</p>

CC BY-SA 3.0
2018-03-04T23:29:00.947Z
@barrycarter You're right. At such small angles it doesn't matter, but still the argument is valid in a strict sense. I'll fix my reply. Thanks!
Florin Andrei
CC BY-SA 3.0
2018-03-05T15:22:56.450Z
@barrycarter oh, my eyes are playing tricks on me; better them than my brain at least. Never mind, thanks, etc.. ;-)
uhoh
CC BY-SA 3.0
2018-03-01T23:34:33.403Z
2018-03-04T23:30:06.157Z
2018-03-04T23:30:06.157Z
Florin Andrei
<p>If i know the radius r of a star in cm, how can I convert it to arcseconds?</p>

<p>For example, if I have a star with r = 3.18e13 cm, and distance to the star d = 220 parsecs, what is the relation to convert the radius from cm to arcseconds. </p>

<p>Thank you.</p>

CC BY-SA 3.0
2018-02-28T17:31:16.637Z
Use trigonometry. $\theta \sim R/D$
ProfRob
CC BY-SA 3.0
2018-02-28T20:05:02.807Z
The definitions of [parsec](https://en.wikipedia.org/wiki/Parsec) and [astronomical unit](https://en.wikipedia.org/wiki/Astronomical_unit) may help.
Mike G
<star><fundamental-astronomy><size><units>
Finding the radius of a star in arcsec
S. Mas
# Finding the radius of a star in arcsec

(Asked by: username_0 on 2018-02-28)

If i know the radius r of a star in cm, how can I convert it to arcseconds?

For example, if I have a star with r = 3.18e13 cm, and distance to the star d = 220 parsecs, what is the relation to convert the radius from cm to arcseconds.

Thank you.

**Tags:** star, fundamental-astronomy, size, units

> **Comments:**
> **username_1**: Use trigonometry. $\theta \sim R/D$  
> **username_2**: The definitions of [parsec](https://en.wikipedia.org/wiki/Parsec) and [astronomical unit](https://en.wikipedia.org/wiki/Astronomical_unit) may help.  

---

## 2 answers

### Answer by username_3 on 2018-03-01. Score: 4

Using basic circular maths: where $d$ is the distance of the star from the observer in AU, and $r$ is the star's radius in AU, and $a$ is the angle encompassed by the radius of the star in degrees:

$
r = \frac{a}{360}2\pi{d}
$

Now rearrange it to make $a$ the subject:

$
a = \frac{180r}{\pi{d}}
$

To get $a$ in arcseconds, you need to multiply the result by 3600 (because there are 3600 arcseconds in a degree):

$
a = \frac{648000r}{\pi{d}}
$

Now, convert your numbers from cm and parsecs to astronomical units:

$r$ = 3.18e13cm = 2.126 AU

$d$ = 220pc = 4.538e7 AU

Put them into the equation:

$
a = \frac{648000\times2.126}{{4.538\times10^7}\pi}
$

To get:

$
a = 0.00966327′′
$

Hope that helps!

**Edit:**

As username_2 has pointed out, the final equation I've ended with can be further simplified if you use different units, to:

$
a = \frac{r}{d}
$

where $r$ is in AU and $d$ is in parsecs, giving an answer $a$ in arcseconds.

This is because the equation $a = \frac{648000r}{\pi{d}}$ has $\frac{648000}{\pi}$ in it - which is the definition of a parsec in AU. Therefore, multiplying $d$ by this number to convert it from AU to parsecs, we get $a = \frac{648000r}{648000/\pi\times\pi{d}}$. Cancel out the pi on the bottom, and cancel the ${648000}$ on the top and bottom, and you're left with $a = \frac{r}{d}$ where $r$ is in au and $d$ is in pc.

> **Comments:**
> **username_2**: I got the same result by using $a_{as} = r_{AU} / d_{pc}$.  
> **username_3**: @username_2 Yes, I wasn't aware there was an actual equation for it; I just used my (limited) knowledge of how circles work to find an equation. I guess your one's more succinct! It makes sense, because the equation $a = \frac{648000r}{\pi{d}}$ has $\frac{648000}{\pi}$ in it - which is the definition of a parsec in AU.  Therefore, multiplying $d$ by this number to convert it to parsecs, we get $a = \frac{648000r}{648000/\pi\times\pi{d}}$. Cancel out the pi on the bottom, and cancel the ${648000}$ on the top and bottom, and you're left with $a = \frac{r}{d}$ where $r$ is in au and $d$ is in pc.  
> **username_4**: `+1` Working things though step-by-step and explaining along the way is a great way to answer, and can be very helpful for the OP or for future readers.  

---

### Answer by username_5 on 2018-03-01. Score: 1

```
2 * arctan(r / d)

```

Use Wolfram Alpha, it takes free-form units like cm, parsecs, etc and usually does the right thing. And you can specify the unit for the answer you're looking for (e.g. arcseconds).

[http://www.wolframalpha.com/input/?i=2+\*+arctan(3.18e13+cm+%2F+220+parsecs)+in+arcseconds](http://www.wolframalpha.com/input/?i=2+*+arctan(3.18e13+cm+%2F+220+parsecs)+in+arcseconds)

In this case, the answer is 0.019 arc seconds

> **Comments:**
> **username_5**: @barrycarter You're right. At such small angles it doesn't matter, but still the argument is valid in a strict sense. I'll fix my reply. Thanks!  
> **username_4**: @barrycarter oh, my eyes are playing tricks on me; better them than my brain at least. Never mind, thanks, etc.. ;-)  

---

