Area of trapezoids

So right here, we have a four-sided figure, or a quadrilateral, where two of the sides are parallel to each other. And so this, by definition, is a trapezoid. And what we want to do is, given the dimensions that they've given us, what is the area of this trapezoid. So let's just think through it. So what would we get if we multiplied this long base 6 times the height 3? So what do we get if we multiply 6 times 3? Well, that would be the area of a rectangle that is 6 units wide and 3 units high. So that would give us the area of a figure that looked like-- let me do it in this pink color. The area of a figure that looked like this would be 6 times 3. So it would give us this entire area right over there. Now, the trapezoid is clearly less than that, but let's just go with the thought experiment. Now, what would happen if we went with 2 times 3? Well, now we'd be finding the area of a rectangle that has a width of 2 and a height of 3. So you could imagine that being this rectangle right over here. So that is this rectangle right over here. So that's the 2 times 3 rectangle. Now, it looks like the area of the trapezoid should be in between these two numbers. Maybe it should be exactly halfway in between, because when you look at the area difference between the two rectangles-- and let me color that in. So this is the area difference on the left-hand side. And this is the area difference on the right-hand side. If we focus on the trapezoid, you see that if we start with the yellow, the smaller rectangle, it reclaims half of the area, half of the difference between the smaller rectangle and the larger one on the left-hand side. It gets exactly half of it on the left-hand side. And it gets half the difference between the smaller and the larger on the right-hand side. So it completely makes sense that the area of the trapezoid, this entire area right over here, should really just be the average. It should exactly be halfway between the areas of the smaller rectangle and the larger rectangle. So let's take the average of those two numbers. It's going to be 6 times 3 plus 2 times 3, all of that over 2. So when you think about an area of a trapezoid, you look at the two bases, the long base and the short base. Multiply each of those times the height, and then you could take the average of them. Or you could also think of it as this is the same thing as 6 plus 2. And I'm just factoring out a 3 here. 6 plus 2 times 3, and then all of that over 2, which is the same thing as-- and I'm just writing it in different ways. These are all different ways to think about it-- 6 plus 2 over 2, and then that times 3. So you could view it as the average of the smaller and larger rectangle. So you multiply each of the bases times the height and then take the average. You could view it as-- well, let's just add up the two base lengths, multiply that times the height, and then divide by 2. Or you could say, hey, let's take the average of the two base lengths and multiply that by 3. And that gives you another interesting way to think about it. If you take the average of these two lengths, 6 plus 2 over 2 is 4. So that would be a width that looks something like-- let me do this in orange. A width of 4 would look something like this. A width of 4 would look something like that, and you're multiplying that times the height. Well, that would be a rectangle like this that is exactly halfway in between the areas of the small and the large rectangle. So these are all equivalent statements. Now let's actually just calculate it. So we could do any of these. 6 times 3 is 18. This is 18 plus 6, over 2. That is 24/2, or 12. You could also do it this way. 6 plus 2 is 8, times 3 is 24, divided by 2 is 12. 6 plus 2 divided by 2 is 4, times 3 is 12. Either way, the area of this trapezoid is 12 square units.