how do you determine the area of a trapazoid without having the height?

Answer 1

you have to have the height otherwise you can't figure it

unless you do something like this:
let's say base 1= 3 and base 2= 4

a= 1/2 (base 1 + base 2)h

you could leave it as

a= 1/2(3)h
and thats your area

or (3h)/2

Answer 2

Depends on the other measurements you have including partial lengths, angles, etc. If you had a couple angles, for example, you could calculate height if you had sufficient other information.

You would need to supply a diagram if you want to figure a specific trapezoid's area.