Macbook Pro overheating

Like many people, I have a MacBook Pro that overheats, but not of the suggested fixes are appropriate for my problem. I live somewhere quite rural and the area is prone to power cuts. I only get overheating when using the Pro at this time. I close down Safari, Chrome and turn off Wi Fi so that it isn't attempting to connect. It makes no difference. Within a few minutes it gets very hot and the fan starts to run. This is also a problem because I'm obviously running on battery (no power available) and the fan depletes the battery quite quickly. The first time I had a Preview image on split screen with an Excel spreadsheet and I was transcribing detail from the image to the spreadsheet. This has now happened several times. Today after it happened again, I let the thing cool down and then started looking at just the spreadsheet with no additional app. The same thing happened. After the first time, I called Apple support and they just said to make sure I had all the updates. I did, updated again, and it's made no difference. BTW I've had it about 6 weeks, brand new. How hot is it safe to get to, and how can I tell what temperature it actually is? It's not too hot to touch, but not far away. It seems ridiculous that I buy equipment specifically made to be portable but doesn't work properly if it needs a power source to work (necessary for the internet)