Areas of Shapes

Alright. So now we're gonna learn how to calculate the area of a triangle on the graph. Um We're gonna do this a few different ways in this class but the formula stays the same. You might remember this area of a triangle equals half times base times height. Uh We usually just write this as half B. H. Right? That's the same thing. Half be H. Um So that is our area of a triangle formula. Let's go ahead and use it on the graph a few different ways. So here on graph a. Um what I wanna do is I'm going to pick this point right here and I want to calculate the area um below the blue line but above that point. All right. So that sounds a little crazy but let's go ahead and visualize it on the graph. It's gonna be above or below the blue line and above that point. So what we would have is this line here connecting this to the point. And now you can kind of see what triangle I might be talking about. I'll highlighted here in yellow. Alright. So there's gonna be a few times in this class that will use a graph like this to calculate this area. Alright. So how do we do it first we have to define what's gonna be our base and what's gonna be our height and then we have to find out what those are. Right? So here we're gonna have our base B. This part right here, the dotted line and our height is gonna be coming up right here. That's gonna be our height. So I'll put an H. I'll put an H out here and I'll leave that little squiggly thing. Okay, so that's gonna be our height. Now. What are those numbers? We gotta figure that out. So for our base let's see what it is. We need to find basically what the change in that X value is. So it looks like we started here at zero for the X values, right? Zero down here and we went all the way to three, right from 0 to 3. So the change there is gonna be three, right? We changed three. So the length of that segment is three, let's do the same thing for the height looks like we started at three and we went up to six. So what's the change there? The six minus the three is gonna tell us that our height is three. Whoops, let me draw that a little better. So our height is equal to three, our base is equal to three, our height is equal to three. So we are ready to do our formula, I'm gonna do up here. Um A for area A for area equals a half times base times height. Our base was three. Our height was three, so half times three times three. That's going to be nine over to right and that simplifies to 4.5 as well. Okay so we are going to do a fractions. Review a decimal. Review all this stuff. If any of this math is tripping you up, we have reviews for all of it. All right, so we're gonna go through all that stuff. Let's go ahead and do um example be here. I'm going to get out of the way and let's do something similar. We've got that same point there where they're intersecting. Um But now I want to go ahead and find the area below the this this dotted line. Okay, so now before we did the area above now we want to do the area below. So again I'll highlighted in yellow here, the area I'm talking about. Right? So this area um highlighted in yellow, this triangle, how do we figure out what the area is? So again we're gonna use our same formula a half base times height. So we need to know what our bases and what our height is. So here we've got a base along this dotted line. Again, the base is going to be there, okay and our height is going to be this change in the y over here, so I'll put it out here. H um And again let me highlight what the height is gonna be. It's gonna be this region right there. It's gonna signify the height and I'll do one over here because I like drawing these squiggly, that's going to be our base. Cool. So let's go ahead and calculate what those are. So it looks like for our base we started here again at zero, right? And it went all the way to four. So from 0 to 4, the change four minus zero, the base is going to have four units. How about the height? Well it looks like we started here at one and it went up to four. So four minus one, that's gonna give us a height of three. So here our base was four, our height was three. We are ready to calculate the area area equals half base times height. So we've got half um times our base of four. Our height of three which is gonna be 12/2, it's going to equal six. Okay so don't get caught up on the math. The idea is how do we use this formula one more example here example c. I've got space on the right so I'm coming back. Hey guys. Alright so example see in this situation I want to find the area of this triangle right here. So they could give us this point right here, they could tell us that the X. Value is two and we need to find the area in between this point. So this one seems a little trickier, right? At least a little bit. How do we calculate this this area right here. Alright so though it seems a little trickier, it is almost the same. Um We gotta find the height. So if you look uh kind of sideways at this triangle, you'll see that we've got the height right here along the middle. You see this this line I just drew in black, That's actually gonna be our height right there. That's the height of that triangle. And our base is gonna be this whole long line right here that I'm gonna do I'll do in in green. So this whole long line right here connecting those two points, that is our base. Cool, so triangle looks a little trickier, but in the end, once we define our base, define our height, the math gets a little easy again. So let's go ahead and see what um what the change in our base and what the change in our height is. And honestly you could you could um do it in any order, right? So let's just start with the base and let's see what the change was. So we started here at zero right zero on the Y axis. And it went all the way up to six. So six minus zero. Our base is going to be six, so base equals six. How about our height? It looks like our height. We started here at two and we went to four. So four minus two. Right? We started their height ends, their four minus two is equal to two. So our height is going to equal to. So now that we know are basis six, our height is two. Let's go ahead and calculate our area area equals half be H. So half times the base of six times the height of two. We've got 12/2. Came out to six again, so the area of that yellow region is six. Cool. That's three different ways that we're gonna use this formula in this class to calculate areas of triangle. Alright, let's move on.