We have an LED strip connected to output 6 and is powered by the Uno board itself. Shouldn't be drawing too much current, only have 10 elements on the LED strip. While this LED strip is connected to the Uno, I noticed that the regulator is getting very hot.
One thing we haven't done yet is to power the LED strips from an external source, which will be necessary since there will be many strips. At the moment, the students are simply trying to debug their program with the small strip.
If you power the Arduino with an external power supply the on board regulator has to dissipate the voltage difference between input and 5VDC at the total current you draw from the board.
As Klaus Warzecha wrote in an answer to another question
The Arduino is just the brain, not the power station.
If the output voltage of your power supply is 12V, then the regulator will be dissipating 12-5V = 7 V at the total current drawn. Total dissipated power for eg. 150mA (just an estimate) would be P = V × I = 7 × 0.150 = 1W The regulator will be pretty hot from dissipating 1W.
Mind you that the current is not limited to 150mA, enough LEDs can easily increase current and power. If the regulator is stressed (often) enough, sooner or later it will fail. It is better practice to power any load directly from the external power supply, taking the higher voltage into account.
A regulator normally get hot when is driven a current near to the maximum that could be driven, and its a normal thing the regulator keep warm.
A normal regulator like LM7805 could work until 125 C°.
If you try to drive more current that it can probably it will burn, so, maybe its time to you change your power supply to an external one to keep safe your Arduino board.
When a circuit is hot, it's very likely that it is drawing too much current. It may be a component that's getting more current than it's designed for, or a PCB track, or a wire.
So, before you do anything else, I strongly suggest that you get a DMM and measure the current drawn by the board when powering the led strip. Google for ways to measure current with a DMM, as it can be tricky if you never done it before.
Then, compare the value to those specified by the voltage regulator datasheet. The current should be within those values shown in the "Electrical Characteristics", and well below those from "Absolute Maximum Ratings".
