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# Area of trapezoids

CCSS.Math:

## Video transcript

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 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 base this long base six times the height three so what do we get if we multiply six times three well that would be the area of a rectangle that is six six units wide and three 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 six times three so it would give us this entire this entire area right over there now the trapezoid is clearly less than that but let's just let's just go with the thought experiment now what would happen if we went with two times three two times three well now we'd be finding the area of a rectangle that has a width of two and a height of three so you could imagine that being this rectangle right over here so that is this rectangle right over here so that's the two times three 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 let me color so this is the area difference on the left hand side and this is the area difference the area difference on the right hand side if we focus on the trapezoid you see that it reclaims if we start with the yellow the smaller rectangle it reclaims half of the area half of the difference between the smaller rectangle of 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 completely makes sense that the area of the trapezoid D 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 all of that over 2 so when you think about an area of a trapezoid you look at you look at the two bases the the long base and the short base take the area of each of or 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 6 plus 2 times 3 times 3 and then all of that 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 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 let's take the average of the 2 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 that would be a width that looks something like we just orange a width of 4 would look something like this with a 4 would look something like that and you're multiplying that times a 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 can do any of these 6 times 3 is 18 this is 18 plus 6 / - 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