i want to save money and make my own filter but all the ones i can find that are really nice you need a powerhead or water pump. all i have is an airpump and thats all i can afford. do you know of any GOOD diy filters that use an airpump?

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"just do it"

Depends on the air pump really and how powerful it is.

Options:
Under gravel filter (best choice)
Sponge filter
Corner filter

I would say the best you can do is an under gravel filter. A lot of people have abandoned these, but for a 20gal tank it should be enough. Just don't over stock it.

What you need is some plastic and a tube or two. Prop it up somehow and drill tons of holes in it. Then you need to drill a hole big enough for one of the tubes(2 holes if you use two tubes) considering your tank is long I would suggest two tubes, but then you either need to split up the air flow or get a second pump. Look up designs online for an under gravel filter so you get the idea.

Also you can try making a corner filter that will be powered by an air pump. Look it up online for help.

I don't see why you don't want to buy a filter though, Ive seen ones in a LFS for less than $20 that would suit your 20gal.

Note: if you want a filter for that 1gal bowl you have then a sponge filter will be more than enough. But I don't recommend keeping a beta in a bowl...