Why do we use height to calculate the area of a triangle? So in a triangle, one of the many ways to calculate area is $\frac{1}{2}$ base $\times $ height.
However, isn't height only used in $3$D objects and isn't a rectangle a $2$D object?
I noticed a lot of people get confuse with using length, width and height.
If I draw a rectangle on a graph with the $x$ and $y$ axis, no $z$ axis, would the width be the $y$ axis and the length the $x$ axis?
Any ideas?
 A: Notice that two people who disagree about which way is up or down, left or right,  forward or backward, could still agree on what the area of a triangle is. The area needs to independent of whatever conventions we use to determine up and down etc. So the 'height' of the triangle can't depend on any choices that we make about these things.
What you are calling the height of the triangle is also called the altitude of the triangle. See the wiki link: https://en.wikipedia.org/wiki/Altitude_(triangle) 
Perhaps it might help you to think of the area of a triangle as being given by the length of base X length of altitude connecting base to the opposite vertex of the triangle.
The above formula makes no reference to up, down, left, right etc
A: As others have pointed out, semantics is tricky.
Suppose that I drew a rectangle that had an extent of $9$ units along the $y$-axis, and an extent of $1$ unit along the $x$-axis.  How many people do you think would call the $1$-unit extent the length of the rectangle, and how many would call it the width?
Making a hard-and-fast rule about what the length, width, and height of an object are is a hopeless enterprise, because some enterprising soul will surely come up with an exception.  One can generalize, vaguely, that "length" usually refers to the greatest extent of an object, but then along comes an oblate spheroid (like an onion) and puts the lie to that idea.
Instead, mathematics makes use of another tactic, from time to time.  It may leave some terms undefined, and rely only on the relationship between those terms.  When we say that the area of a triangle is one-half the base times the altitude, what we really mean to say is that its area is one-half the length of a base times the length of the corresponding altitude.
We then say that the base is any side of the triangle, and the altitude is the length of a perpendicular dropped from the remaining vertex (not on the base) to the base (or the base extended, if need be).  The area of the triangle, so calculated, remains unchanged no matter which side is used as the base (as of course it must).  But once a side has been chosen as the base, the altitude must be constructed corresponding to that base.  One cannot use, in triangle $\triangle ABC$, the side $\overline{AB}$ as the base, and then drop an altitude from $A$ to side $\overline{BC}$, and use that in the formula.
This may sound trivial, in this instance, but the semantic quibbles are often more subtle than that, and real mathematics may depend on them.  The foundations of arithmetic on set theory, for example, were an attempt to quash many such quibbles.
