Areas of triangles
- [Voiceover] 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, then 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 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 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 gonna try to think about what's the area of this triangle going to be, and you can imagine it's going to be dependent on base and height. Well, to think about that, let me copy and paste this triangle. So let me copy, and then let me paste it, and what I'm gonna do is, so now I have two of the triangles, so this is now going to be twice the area, and I'm gonna rotate it around, I'm gonna 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, 'cause 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 the parallelogram. Now why is this interesting? Well, the area of the entire parallelogram, the area of the entire parallelogram is going to be the length of this base times this height. You also have height written with the "h" upside down over here. It'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 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, "OK, maybe it worked for this triangle, "but I wanna see it work for more triangles." And so, to help you there, I've added another triangle right over here, you could do this as an obtuse triangle, this angle right over here is greater than 90 degrees, but I'm gonna do the same trick. We have the base, and then we have the height. Here, you can think of, if you start at this point right over here, and if you drop a ball, the length that the ball goes, if you had a string here, to kind of get to the ground level, you could view this as the ground level right over there, that that's going to be the height, it's not sitting in the triangle like we saw last time, but it's still the height of the triangle. If this was a building of some kind, you'd 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 magic here. So let me copy and paste this, so I'm gonna copy and then paste it. Whoops, that didn't work. Let me copy, and then paste it. And so, I have two of these triangles now, but I'm gonna flip this one over, so that I can construct a parallelogram. So I'm gonna flip it over, and move it over here, I'm gonna have 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 twice the area of our original triangle. And so, if I talked about the area of the entire parallelogram, 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 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 gonna 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 paralleogram is going to be base times height.