My internal storage appears to be almost used up but the math says there's not that much there

[gwhale2005]

Adding up the downloaded apps. And the memory that the phone takes to run. The total of the two combined plus remainder don't add up to 16gb

 

[webworkings]

You are forgetting the preinstalled apps and the OS for the phone itself. Those take up tons of space. Plus, any photos, videos, etc. also use space.

 

[bahnstormer]

I find that something like "MyDrive Storage Analyser" by Anton Patapovich is really decent as it allows you to drill into different groups of folders and see the file types, etc.... it was amazing when I was trying to keep my head above water on my old Galaxy S2 (also 8Gb) and it allowed me to see that I had accidentally run Titanium Backup to backup the whole OS partition and then stored it on the internal storage! I used it again when I was running to the end of life with my Galaxy S4 and waiting for a suitable replacement to come to market (Samsung and most other manufacturers had dropped the IR blaster, removable battery, etc).

I can't recall what the limitations were on the Free version, but there is one and I used that for a while before deciding to pay the developer for his time and effort. Pretty sure the Free one will be plenty for your needs.

edit: Play Store link (to paid version): https/play.google.com/store/apps/details?id=com.mobile_infographics_tools.mydrive_ext

And Free version: https/play.google.com/store/apps/details?id=com.mobile_infographics_tools.mydrive