Why are break-even and shut down points based on average cost? I understand the basic idea of break-even and shut-down points, where break-even is the price at which revenue covers all economic costs, and is located where the marginal cost equals to average cost. And that shut down is the price where revenue just covers variable costs (it is where marginal cost equals average variable cost).

My question, why are taking in to account average cost and average variable cost, rather than just total cost and just variable cost? Why is shut down point where revenue equals average variable cost rather than just variable cost or total cost, and why is break even where marginal cost equals average total cost, not just total cost or variable cost?

Thank you
When you are finding the break even point, you are looking for a price-point.

The price is essentially the average (and marginal) revenue. Suppose price of good $x$ is $p$.

Revenue = $p * x$.

Average Revenue = Revenue / $x$ = $p$.

So what you are really doing when looking for p is to set Average Revenue = Average Cost. 

At that point, you will also have the fact that Revenue  = Cost. So your logic carries through. You still break even at the point where revenues exactly cover costs.

To see this, note Average Costs = Costs / $x$.

So Average Costs = Average Revenue $\iff$ Costs / $x$ = Revenue / $x$  $\iff$  Costs = Revenue

The interpretation when focusing on average costs is only slightly different. When talking about averages, we're just talking about prices and costs per unit. So you break even at the point where your revenue per unit equals your cost per unit. It is necessarily implied that you also break even where your total revenue equals your total cost.