Area of triangles
Current time:0:00Total duration:10:23
Area of triangle proof
We now know how to find the area of rectangles. What I want to do in this video, is think about how we can find the areas of triangles. So, we're starting here with a right triangle, has a 90 degree angle right over here. Right triangle ABC. And let's think about how we can find this area. Well, maybe we can construct a rectangle out of triangle ABC. And if we can construct a rectangle out if it, and then, maybe we can somehow find our area of part of that rectangle. And the best way to construct a rectangle is to really duplicate ABC, and then flip it over and put it right on top of this. And just to verify, that definitely will be a rectangle. So we know that this is 90 degrees right over here. Let's say that this is x degrees right over there. We have x + 90 + this thing have to be equal to 180. So, this thing and this thing have to add up to 90. Let's just call this 90-x. Now, let's flip this thing and rotate it around, so that it will look like this. So, then you would have another triangle that would look just like this. Where now, this right angle, on the flipped version is that right angle right over there. This angle right over here, this x, is now this angle right over there. And this 90-x is now this angle, right over there. And you can see x + 90-x, that'll give you a right angle. And then you have x + 90-x, that gives you a right angle. And you have four sides and four right angles, you're definitely dealing with a rectangle. And this rectangle has two of our original triangles in it. So, we can write, we can write that the area of triangle ABC, the area of triangle ABC, that's what the brackets mean, area of triangle ABC, is going to be equal to ? * the area of our entire rectangle. And our entire rectangle, let me add another point here. Let me call this D, is going to be ? the area of rectangle ABCD. And we know how to find the area of rectangle ABCD. It's going to be equal to, it's going to be equal to the base of the rectangle. So, this is going to be equal to ? * and so this part right over here. Let me just use a different color. The area of ABCD is equal to the base or the width of the rectangle. So, that's just the length of BC. I'm just putting this in parentheses, it doesn't meanďź I just, BC is just the length of this segment right over here I'm just putting in the parentheses so that we don't get the letters jumbled up. So it's going to be this width or this base right here times the height of the rectangle. So, times AD. Timesďź sorry, times AB. So, it's this base * this height gives us the area of the entire rectangle. And the area of a right triangle is half of that. So, there we have it. It's ? * this base * this height. It's the area of a right triangle. So, in general if you ever have a right triangle, if you ever have a right triangle, and this is a right angle right over there which is, you need one right angle in order for it to be a right triangle. And this base has length B, and this side over here has length H. You know that the area, the area is going to be equal to ? * the base of the triangle. That's the base of the triangle BC * the height of the triangle. So you can view it this way, if you look at the actual letter points. And if you just use these measures as a base * height, it's just ? base * height. And we only that this works right now for a right triangle. Now, let's think about it for other types of triangles that aren't necessarily right triangles. So, here I have a kind of an arbitrary triangle ABC. And to approach figuring out its area, what I wanna do is just split this up into two right triangles. So what I'm gonna do is draw a perpendicular from B, so I'm just gonna really, if this was, if this was an actual structure, you'll just literally drop something straight down from here, and that line is going to be perpendicular to this base right over here, to AC. And let me call that point, let me call that point D. And what's useful here is now we've constructed, we've turned that one triangle into two right triangles. So we can say that the area, let me write it this way, we can say the area of triangle ABC, that's what we wanna figure out. It's equal to the area of this character right here. So it's equal to the area of triangle ABD + the area of triangle, + the area of this magenta triangle. So, plus the area of BCD, of BCD. And this is useful because we know how to find the area of right triangles. Now, obviously this is 90 degrees and this is also going to be 90 degrees. The area of ABD is ? base * height. So it's going to be ? * the base, which is the length AD. So ? * AD, * the height, which is the length of BD right over here, assuming that we can figure that out. So this * that length, so, BD. So that's the area of the blue triangle, and I'll find the area of the magenta triangle. Well, once again it's a right triangle. It's gonna be ?, let me do that in magenta. Let me do that in magenta. So it's going to be ? * the length of this base right over here, which is DC, the length of segment DC * the length of BD again. Now, you can factor out a ?, we can factor out a ? BD from both of these terms, ? BD. So you get ? BD * AD, we're left with AD, and that's not the same shade of blue. We're left with AD + DC. We close the parenthesis. And what is AD + DC? AD is the length, that length. And then DC is this length. So if you will add those two lengths, you get the length of AC. So this whole thing, this whole thing is the length AC. So, we're left with the area of ABC is equal to, I'll use in a new color, it's equal to ?, and I'll just switch the orders right over here. ? *, which I'm gonna write in that same color, * AC * BD. Now, what is this again? Well, this is now ? * the base, which is this AC, * the height, which is BD. So that's pretty cool. It worked for right triangles, and it actually, if we know the height of a triangle. Now, this isn't one of the sides now. For a right triangle, it was one of the sides. Now, for this arbitrary triangle, it isn't. But if we know it, the area of this triangle is still ? * the base * the height. Now, what about a triangle like this? How can we figure out its area? Well, let's try doing the same thing. Let's see if we can somehow, either construct this out of right triangles, or maybe out a right triangle of this to make it into another right triangle. And the easiest way to do that is, we could kind of just drop a rock from right over there. And then this, and then where it kind of hits the ground would form a right angle. And let's call this point D. And what we care about, what we wanna find in this, what we wanna find the area of, we wanna find the area of triangle ABC. So that's what we care about. Triangle ABC is what we wanna find the area. But the area of triangle ABC is gonna be the area of this larger right triangle that we've set up. That's gonna be this larger triangle. Minus the area of this smaller one right over here. So, minus the area of ADC, of ADC. So I just, this blue one ADB, this is just the whole thing. This is the whole thing, just so we're clear of what we're talking about. Now, what is the area of ADB. Well, we know how to find the area of right triangles. ADB, the area of ADB is going to be, is going to be ? * our base, which is DB, the length of segment DB * our height, which is the length of AD. Times AD. And then from that we wanna subtract the area of the smaller triangle. So that's gonne be ? * our base, which is DC, that's the length of our base, DC, * the length of our height, which is AD, AD. So, here we can factor out a ?, and we can factor an AD. A ?, and an AD, so let's do that. So we factor out a ? * AD. And what's left inside is a DB - a DC. Let me put the parenthesis in a white again. Well, if you take the length, DB is the length of this whole thing. You subtract from that the length DC, you're gonna be left with CB. So this character right over here, this is C, this is CB. And so the area of ABC is going to be equal to ? * CB, I'm just switching the order of multiplication, with a yellow color. ? * CB * AD Now, what is this? Once again, it's ? * our base * our height. And once again the height in this case, 'cause it's not a right triangle, it's not one of the sides. Someone would have to give you that information. You'd have to figure out what this actual height is. But what's needed is, in any form of triangle the area is ? base * height. The height is, kind of, if you have the sides of a right triangle, it's easy to figure out. For these others, if it's not given, you might have to do it in some tricky way, figure out the height somehow. But anyway, I hope that you find that useful.