Tall building sighting distance

[Echoing Stars]

From what distance can a person on the ground see the Burj Khalifa?

Or: How tall does the building have to be to be seen from 250 kilometers away?

 

[fresh_42]

$4.9 \,\text{km}$

 

[russ_watters]

[Edit] two questions answered....here's the general answer:

https://en.m.wikipedia.org/wiki/Horizon

https://www.ringbell.co.uk/info/hdist.htm

 

[Echoing Stars]

What you said: The building needs to be 4.9 km high to be visible from 250 km away?
I would like an explanation of the equation that produced this result.

 

[fresh_42]

What you said: The building needs to be 4.9 km high to be visible from 250 km away?
I would like an explanation of the equation that produced this result.

See. And I would have liked to have seen some effort from your side to solve the problem.

 

[fresh_42]

That is incorrect:

https://en.m.wikipedia.org/wiki/Horizon

I scribbled it on a tiny kid's toy so I may have made a mistake, but I don't think so.

I used the polar Earth diameter from Wikipedia, $12713\,\text{km}$ and the distance $250\,\text{km}$ as line of sight. I reviewed it, and it is a little more than $4.9\,\text{km}.$

Spoiler

Solve: $0=(r+h)^2-r^2-(250)^2$

 

[Echoing Stars]

You are a champion.
This is my equation:
h=63712+2502−6371h = \sqrt{6371^2 + 250^2} - 6371

 

[fresh_42]

That is incorrect:

https://en.m.wikipedia.org/wiki/Horizon

I calculated it now with more precision:
Earth radius on 25°N (Dubai) = 6374.344 km
(from https://rechneronline.de/erdradius/)
Distance 250 km
Result 4.90058148 km

So my initial approximation of 4.9 km was even way better than I thought.

 

[fresh_42]

You are a champion.
This is my equation:
h=63712+2502−6371h = \sqrt{6371^2 + 250^2} - 6371

I think you have been a bit sloppy in the first equation. The same formula (Pythagoras) also allows you to solve for the distance $x$ if you use the radius 6374.344 km and height (peak) 0.8298 km of the actual building. Then $x=102.857\,\text{km}.$

 

[Echoing Stars]

אני חושב שהיית קצת מרושל במשוואה הראשונה. אותה נוסחה (Pythagoras) מאפשרת לך גם לפתור את המרחק $x$ אם אתה משתמש ברדיוס 6374.344 ק"מ ובגובה (שיא) 0.8298 ק"מ של הבניין בפועל. ואז $x=102.857\,\text{km}.$

And it's just like that..
Look here

 

[russ_watters]

I scribbled it on a tiny kid's toy so I may have made a mistake, but I don't think so.

I used the polar Earth diameter from Wikipedia, $12713\,\text{km}$ and the distance $250\,\text{km}$ as line of sight. I reviewed it, and it is a little more than $4.9\,\text{km}.$

Spoiler

Solve: $0=(r+h)^2-r^2-(250)^2$

Sorry, there were two questions asked: I thought you were answering the first and I skimmed past the second.