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# Area of a triangle on a grid

## Video transcript

what I would like to do is find the area of this green triangle and if you get so inspired I encourage you to get inspired pause the video and see if you can figure it out on your own so whenever you start thinking about areas of triangle or at least my brain says well look I can figure out the area of a triangle if I know the base and the height of the triangle I can just multiply them and then multiply that by one half so for example if I have a triangle that looks like this a triangle that looks like this and if if this is the base this is B right over here the length of this side is B let me put the B in magenta and then the height let me do that in yellow this is the height this is the height then I just multiply base times height times one-half and I'll get the area of this triangle or if the triangle look like this if it looked like if it looked like this if it looked like this I could do the same thing where if this is the base B of that is the base B and now the height I guess you could say this if you're to drop a penny from here it's sitting outside the triangle so it looks different than this one but this would still be the height this would still be the height right over here you do the same thing 1/2 times base times height would give you the area of this triangle so how can we apply that over here well this triangle is on this grid but it's kind of at an angle with this grid it's hard to pick out the base and the height for for this triangle as a whole but what we could do there's actually several ways that we can approach this is we can break this triangle up into two or more triangles where we can figure out the base and the height for each of them so for example I can break this one let's see I could I'm picking this point in this point because it breaks it up into two triangles where I can figure out the base and the height well what am I talking about well this triangle over here that I am shading in blue if I it and I've switched the orientation I've rotated 90 degrees but if you view this yellow if you view this yellow as the base of this triangle you see that the base is 3 so let me write the base is equal to three units and what's the height here well the height here is going to be the height here is going to be this distance right over here which is four height is equal to four so the area the area of that triangle right over there is going to be one half times three times four which is equal to six so this part right over here the area is six and now we can do a similar thing with this other triangle because once again we can view this yellow line or now if this yellow and blue line as the base the base is equal to three so I could write that base is equal to three and once again I've rotated so now the base is on the side so the base is three and then the height here the height of this triangle is two if this is the base remember if this is the base here we've just rotated it then this right over here is the height height would be equal to two so what's the area of this one the area of this one is going to be one half times the base three times the height which is 2 1/2 times 2 is 1 times 3 this is going to be equal to 3 so the area of the whole thing is going to be this area of 3 plus this area of 6 it's going to be an area of 9 area is equal to 9 now that's one way you could do it is you could break it up into triangles where you could figure out the base in the height another way and this is you can kind of view it as a maybe a trickier way or you kind of have to think a little bit outside of the box or maybe outside of the triangle to do it this way is to instead of doing it this way visualize this triangle and actually let me let me clean this up a little bit let me undo all this work that I just did let me undo this too to show you the other approach the other way that we could tackle this so the other way we could tackle it I'm going to clean up the whole thing so I get more so I get more real estate here so the other way that we could tackle it is imagine that this try this triangle is embedded inside of a rectangle so let me draw the rectangle let me draw let me draw the broader rectangle and I think you might see where this is going because as soon as you draw that that bigger rectangle then you see that that rectangle is made up of the triangle board that we're trying to find the area of and three other right triangles we have this right triangle that I'm shading in in yellow we have this right triangle that I'm shading in in purple and then we have this right triangle that I am shading in in blue so if we figure out the area of the entire rectangle and that's pretty straightforward the area of the entire rectangle is going to be four times six those are the dimensions of the rectangle four times six so the area the entire rectangle is 24 and then you subtract out the area of the purple the blue and the yellow rectangle the purple the blue and the yellow triangles then you're going to be left with the area of the green triangle so let's do that so what's the area of the purple one well this is going to be we're going to subtract it out it's going to be one half six this its height right over here if we view this as a side of six and its base right over here is three so it's going to be 1/2 times six times three that's the area of the purple triangle and then you have the blue one this is going to be minus 1/2 let's see you could say its height is 1 so this is one and then this base you could say is four so that's four times four and then we want to subtract out the area of the yellow rectangle so this is minus 1/2 let's see if we can make this base 2 1/2 times 2 times and this height is 4 2 times 4 so what's this going to be well let's see 1/2 times 6 times 3 that's 3 times 3 that's going to be 9 1/2 times 1 times 4 that's going to be 2 and then 1/2 times 2 times 4 well that's just going to be 4 and so we're left with 24 minus 9 minus 2 minus 4 so this gets us let me do the same color minus 4 so what is that 24 minus 9 is Dean 15 minus 2 is 13 13 minus 4 is equal to 9 it's equal to 9 so that's the other way or another way to get the area of this green triangle