Let's tackle the case of a triangle, where one only needs to envision shortening one of the trapezoid's parallel sides down to a single point. For the sake of convenience let us imagine that the side being shrunk is the side labeled "d" (i.e., the top-side) as shown in the illustration below... This side is then continuously shortened until it diminishes to a point.  Which now means that the length "d" becomes "0" (width of a point). The algebraic result, below, entails the following where "0" is substituted for "d" in both trapezoid formulas:

P =	a + b + c + 0	&	A =	0.5(b + 0) × h
=	a + b + c		=	0.5(b) × h
			=	0.5bh
			or	bh ÷ 2

Voilà!  Each of the formulas for a triangle easily appear right out of the two trapezoid formulas. Interestingly enough, we shall soon see a similar feat occurs with both a rectangle and a square.  Onward then to how this works for the rectangle.

     In the instance of a rectangle, the trapezoid needs to transform where both parallel sides are equal in length and coincide with each other in horizontal position. Once more for convenience's sake we choose to alter the side labeled "d" (top-side in the illustration) allowing it to stretch until it reaches a length which is identical to the base (bottom-side) labeled "b."  And yet another key observation (just prior to proceeding with the analogous algebraic effects) is to take note that both non-parallel sides labeled "a" & "c" are perpendicular to the base and are equal in length to each other (i.e., a = c) — in fact, they are both equal to "h" (height of the figure).
     Where does this leave us?  We can utilize the equivalences "a = c" and "b = d," in both of the trapezoid formulas as follows:

P =	a + b + a + b	&	A =	0.5(b + b) × h
=	2a + 2b		=	0.5(2b) × h
=	2h + 2b		=	1b × h
=	2b + 2h		=	bh