Jennifer, Welcome to the forum! You can use this to accurately measure the surface area of your pond:
http://acme.com/planimeter/ When used correctly, the chemicals are only harmful to the plants and algae that they are supposed to harm. But I can understand you not wanting to use any.
Without using ANY chemicals, you have to be resigned to mechanical removal of the algae - i.e. raking it out. Algae is caused by an excess of nutrients in the pond, and the only thing that will help that is a lot of plants growing in the pond. Clear water will let sunlight penetrate further into the water, allowing plants and algae to grow in progressively deeper water.
You can look for Phycomycin or Green Clean Plus. Basically it's a very strong form of hydrogen peroxide. But that is a chemical compound.....
If you do not want plants or algae in the pond, you will have green colored water. That's actually good - it is phytoplankton, which is the bottom of the food pyramid in the pond. It limits sunlight getting to the pond bottom, and helps limit plant growth.
But, you want clear water, so that means having a lot of plants growing in the pond to use the nutrients before the algae and phytoplankton can.
I would look into bacteria and aeration (if the pond is deep enough to set up a thermocline) to help take care of the muck on the bottom of the pond. You can get a blend of bacteria that is specifically tailored to your pond by calling these guys. You will need to send a sample of your muck to them, but they can walk you through the process to do that.
https://teamaquafix.com/The ONLY way to have clean, clear water in the pond without using any chemicals and without having plants in the pond is to dig another pond the same size as that one, fill it full of submerged and emergent plants, and pump the water from the original pond through it, using the 2nd pond as a biological filter. But, that still won't help you with the existing muck problem. Either mechanical cleaning out of the pond or weekly/bi-weekly bacterial treatments will do that.