Why is the price of capital ‘r’ ? (From Cost function) according to the Cost formula in microeconomics class,

Total Fixed Cost is represented as “rK” (K as in unchanging, fixed K) 

now my economics teacher tells me this ‘r’ is the interest rate at which you rent capital, which is pretty vague and ambiguous to me. If r represents interest “rate” and not the “interest-added cost of capital”, then how can this r be the “price” of capital?

So for example, if the amount of capital is 5 (K=5) I understand that you assume you rent it, or borrow the money from a bank to get the capital, incurring interest. What I don’t understand is the formula. If r is the interest rate (aka rental rate) 

isn’t the cost formula supposed to be “K * {P(K) + P(K)*r}”, instead of “rK”?
 (P(K) here represents the price of capital per unit) 

so if a unit of capital costs $10, and the interest rate (r) is 3%, 

in the end you will be paying $ 5 * (10 + 10 * 0.03) in total , not 5 * 0.03 right? 

I mean, r is just the percentge, not the ‘amount’ that you have to pay for the capital. So I don’t really understand why the textbooks and the teachers tell me that the “price” of capital is r when it seems like it’s supposed to be ‘P(K) * (1+r)’ instead of just ‘r’ . 

I hope I can get a satisfying answer here. Would appreciate detailed explanations!
Firstly, in your example the value of $r$ (as used by economists in this context) would be $1.03$, not $0.03$. Economists call this the "interest rate", but you might prefer to think of it as the "rate of return on capital".

Secondly, what we define as constituting one unit of capital is pretty arbitrary. Is a computer one unit of capital or ten units of capital? It doesn't make any difference, just like it doesn't matter whether you measure distance in meters or feet, so long as everyone knows what units you are using. Since this choice is arbitrary, economists usually make the simplest possible choice and measure units of capital in units of money (e.g., dollars), meaning that one unit of capital has a price of one dollar by definition: $P(K)=1$.

If we return to your formula with $r=1.03$ and $P(K)=1$:

$$K[P(K)+0.03P(K)]=P(K)K(1+0.03)=Kr,$$

we get exactly the cost you saw in your course.