I'm in a fellowship that pays for airfare for a weekend event (fly in Friday, fly out Sunday). Since I'm flying across the country, I thought it might be nice to extend the stay, book another hotel after the event, and take a couple days for myself.
Is it rude to ask the organization paying for the flights and my stay if I could instead fly out a few days later? They haven't booked anything yet or asked for travel details.
In general it would not be considered rude to do what you're asking, however I would make a point to specifically state that you will cover any additional costs such as the hotel, and that you are not asking them to cover anything extra as a part of your request.
The most likely point of contention would be that the airfares may be more expensive for the dates you want to stay, and the company may not be willing to cover the additional charge, and may not have a way to split the costs with you even if you are willing to pay for the additional charges.
As long as you make it clear that you are requesting this as a favor, and be understanding if they refuse, then I would certainly suggest making the request!
Any organization that regularly books flights has probably encountered this question before and may well have a policy to address it. Rudeness is a cultural question, and you haven't specified where you are, but it wouldn't generally be rude to ask this in the United States.
One key thing to keep in mind is that travel prices vary, often significantly, depending on the travel dates. The organization likely doesn't want to incur any additional costs for your vacation, so they may only be willing to pay for flights as if you're traveling Friday-Sunday, if the weekday flights are more expensive, with you paying the difference. That, in turn, may create accounting headaches they don't want to deal with.
Some companies do have policies against doing this (insurance/liability reasons are sometimes mentioned, or just a policy to simplify the planning and accounting). But I don't think it would be at all rude to ask as long as you are clear that you'll make it cost-neutral for them and would understand if this isn't possible.
