New Harddrive

Hello,
I got a used laptop that needs a new hard drive. I found the exact model number that's already in the laptop I just need to buy it. I'm wondering how to reinstall the original OS on the new hard drive.
The new hard drive says it's not formatted. I was thinking about ordering the recovery disc from the manufacture website for my exact laptop model, However it's almost $30 for these disc and I'm not sure if a "Recovery Disc" would work or not. Was wondering would this website work https://wudt.codeplex.com/ ? If I created a bootable usb maybe I don't need the disc?

EDIT: Do I need to format the drive before attempting to reinstall the OS and how? Or will the method of reinstalling the OS automatically format the drive?

First of all, you don't need to order the exact same model hard drive. The same model will work, of course, but you could also take this as an opportunity to install a larger capacity drive (or even a smaller) or even switch to an ssd instead of an hdd. Unless your laptop is more than a few years old, a 2.5" SATA drive of any capacity should work.

The tool you linked to requires you to first have an ISO of an OS disk in order to write it to a USB and make a bootable USB from it. Does the laptop have a Windows sticker on the bottom with a Product Key? If so, you may be able to download an ISO of the OS.

Your best best is probably to order the recovery disk from the manufacturer. Not only will this install the OS, but it will also install all the required drivers and auxiliary programs. Unfortunately, it will also install any bloatware that came from the factory as well.

You do not need to format the drive beforehand.