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CCSS.Math:

we know that we can find the area of a rectangle by multiplying the base times the height the area of a rectangle is equal to base times height in another video we saw that if we're looking at the area of a parallelogram and we also know the length of a base and we know its height that the area is still going to be base times height now it's not as obvious when you look at the parallelogram but in that video we did a little we did a little manipulation of the area we said hey let's take this little section right over here so we took that little section right over there and then we move it over to the right-hand side and just like that you see that as long as the base and the height is the same as this rectangle here I'm able to construct the same rectangle by moving that area over and that's why the area of this parallelogram is base times height I didn't add or take away area I just shifted area from the left-hand side to the right-hand side to show you that the area of that parallelogram was the same as this area of the rectangle it's still going to be base times height so hopefully that convinces you that convinces you that the area of a parallelogram is base times height because we're now going to use that to get the intuition for the area of a triangle so let's look at some triangles here so that is a triangle and we're given the base and the height and we're going to try to think about well what's what's this area what's the area of this triangle going to be and you can imagine it's going to be dependent on base and height I want to think about that let me let me copy and paste this triangle so let me copy and then let me paste it and what I'm going to do is so now I have two of the triangles so this is now going to be twice area and I'm going I'm going to rotate it around I'm going to rotate it around like that rotate it around like that and then add it to the original area and you see something very interesting is happening I have now constructed a parallelogram I have now constructed a parallelogram that has twice the area of our original triangle because I have two of our original triangles right over here you saw me do it I copied and pasted it and then I flipped it over and I constructed and I constructed the parallelogram now why is this interesting well the area of the entire pallet parallelogram the area of the entire parallelogram is going to be the length of this base base times this height you also have the height written when the H upside-down over here what's going to be base times height that's going to be for the parallelogram for the entire let me draw a parallelogram right over here that's going to be the area of the entire parallelogram so what would be the area of our original triangle what would be what would be the area of our original triangle well we already saw that this area of the parallelogram it's twice the area of our original triangle so our original triangle is just going to have half the area so this area right over here is going to be one-half the area of the parallelogram one half base let me do those same colors one half base times height one half base times height and you might say okay maybe it worked for this triangle but I want to see it work for more triangles and so to help you there I've added another triangle right over here you could view this as an obtuse triangle this angle right over here is greater than 90 degrees but I'm going to do the same trick we have the base and then we have the height here you can think of if you start at this if you start at this point right over here and if you drop a ball it the length that the ball goes or if you had a string here to kind of get to the ground level you could view this as the ground level right over here that's going to be the height it's not sitting in the triangle like we saw last time but there's still the height of the triangle this was a building of some kind you say well this is the height how far off the ground is it well what's the area of this going to be well you can imagine it's going to be one half base times height how do we feel good about that well let's do the same let's do the same magic here so let me let me copy and paste this so I'm going to copy and then paste it whoops that didn't work let me so let me copy and then paste it and so I have two of these triangles now but I'm going to flip this one over so that I can construct a parallelogram so I'm going to flip it over and move it over here I'm going to rotate it a little bit more so I think you get the general idea so now I have constructed a parallelogram that has twice the area of our original triangle it has it has twice the area of our original triangle and so if I took about the area of the entire parallelogram it would be it would be base times the height of the parallelogram base times the height of the parallelogram but if we're only talking about the area of if we're only if we're only talking about this area right over here which is our original triangle it's going to be half the area of the parallelogram so it's going to be one-half of that so our area of our original triangle is one-half base times height so hopefully that makes you feel pretty good about this formula that you will see in geometry that area of a triangle is one-half base times height while the area of a rectangle or a parallelogram is going to be base times height