Hello to everyone and Merry Christmas.
I currently have the stock intel cooler that came with i3 3225 and I've found that the noise from the CPU cooler is a big problem, not temp very cool. The cpu is placed inside a Matx case (midi).

Can you give me some advice on a cooler (with price range of euro 20-40) that would be as quiet as possible, while keeping things cool!
Thanks

Hi and welcome to SPCR. Without knowing the specific case (and it's cooler height limitation) it's hard to give you a suggestion.

A very cheap option would be the Arctic Cooling Alpine 11. It has the same thermal performance as a stock intel cooler, but it is much more quiet! It is practically silent in my built when cooling a triple core 95 W AMD Athlon II 425.

The case is from an old compaq pc sr1279it. The case is large 17 cm, free space inside about 15 cm: see attached photo.
Kuzzia thanks for suggestion. Can you specify which Alpine 11? Plus, pro, gt? In the meanwhile I've examined another Arctic cooler: Freezer 11 lp, what do you think?
Ciaoo

Well, your CPU is a 55 W CPU i.e. it doesn't dissipate much heat. If you don't push it a lot the GT version should suffice. It only has a 80 mm fan opposed to the other two which have 92 mm fans. If you game the additional cooling performance of the Pro version becomes beneficial.

The min. fan speed:

GT: 500 rpm
"vanilla": 900 rpm
Pro: 500 rpm
Freezer 11 LP: 900 rpm

The noise at 900 rpm is audible so I'd only recommend the ones with 500 rpm.

Some other possibilities:

Scythe Samarai ZZ Rev B ~$30
Scythe Big Shuriken 2 Rev B ~$40
Thermalright TRUE Spirit 120M ~$43