I'll do you one better. I'll show you how it's derived!
Suppose we have a completely RANDOM trapezoid. I've chosen
this one at random.
Now, let's do what you did. Let's split it up into 2 triangles and a rectangle (we can't assume height is the same size as length, so it's not a square necessarily) by drawing vertical lines.
So our left triangle - it has height
h, and I'm going to call its length
x.
Our right triangle has the same height, but is shorter - I'll call it length
yNow our rectangle also has height, h, but I'll call this length
b1The entire bottom side of our trapezoid MUST be the lengths of the two triangles plus the length of the square, right? So I'll call this longer length
b2 then
b2=b1+x+yNow we know the area of a triangle, it's 1/2 base x height, yeah? So we have two different bases: x and y.
Area of the two triangles must be:
21xh+21yh, and our rectangle area is simply
b1hSo to get the total area of the trapezoid, we add them all together like you did in the exam:
A=21xh+21yh+b1hThis is the area of a trapezoid, but it's not very pretty is it? Let's fix that: Take out h as a common factor, since it's in all of them!
A=h(21x+21y+b1)Now the next step is trickier to see, but we can actually take out 1/2 as a common factor, let's see if you see the trick:
A=21h(x+y+2b1) - I've taken a half out as a factor of each of the terms, which means I have to double everything inside the bracket to keep it correct. You can see that this is true simply by expanding the bracket out - you get what you had at the start again!
Now the next step is easy:
A=21h(x+y+b1+b1) - we've just rewritten 2b_1 is all. You know, like 2x = x + x, same deal here.
Remember what we said at the start? We said that the longest side of the trapezoid is
b2=x+y+b1. We have that inside the bracket! So let's substitute it in, yeah?
A=21h(b1+b2). And there we go. That's the area of a trapezoid by using the length of the two parallel lines, but we can make it a little bit prettier by moving things around a little and putting the 1/2 back in the bracket:
A=(2b1+b2)hTADA! The area of a trapezoid as you were taught at standard grade!
You don't actually have to remember very many "area of..." formulas. You could honestly just remember the area of a square - and then you can prove the area of a lot of other shapes (the triangle, then the trapezoid for example) from that, as long as you can take some simple little logical steps! I had no idea what the area of a trapezium was, but just decided I'd work it out there.