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## Geometry (all content)

### Course: Geometry (all content) > Unit 7

Lesson 6: Area of shapes on grids# Area of a triangle on a grid

Learn how to calculate the area of a triangle on a grid. Created by Sal Khan.

## Want to join the conversation?

- I'm not sure I understand how i'm supposed to find the area of the triangle. Can some one help me PLZ.(8 votes)
- the height times width then divide it in two(18 votes)

- I always thought that you were supposed to multiply the base and height, and then you divide that number by two. Is that still a good method?(9 votes)
- Yes that is how my teacher tought me yesterday(2 votes)

- If you multiply 1/2 to the base and the height What do you do?(2 votes)
- How do you know what the base or height is? Couldn't it be any side?(3 votes)
- Yes, it can be any side you choose. Just be sure that the base is
*perpendicular*to the height: b⟂h(4 votes)

- What if you didn't have the height or the area?(3 votes)
- Then you split the triangle into two or more triangles and find the height and base of each of those triangles. Then you can find the area(2 votes)

- Can the following formula be used to find the area of a triangle on a grid, given the coordinates?
*area = Ax(By-Cy)+Bx(Ay-Cy)+Cx(By-Ay)/2*

Ax = x coordinate for point A

Ay = y coordinate for point A

Bx = x coordinate for point B

By = y coordinate for point B

Cx = x coordinate for point C

Cy = y coordinate for point C

https://www.mathopenref.com/coordtrianglearea.html(3 votes) - The second approach Sal takes only works because all of the vertexes of the triangle fall exactly on the edge of a unit box, right? Would you be able to use this method if the edges of the triangle weren't so exact?(2 votes)
- You may have to use Pythagorean Theorem or trig functions to find lengths of different sides of triangles along with possibly having to find where a line is perpendicular to another line and goes through a given point.(3 votes)

- the pink triangle is exact copy of the shaded triangle so we just need to take out the area of pink triangle.(3 votes)
- I don't get it. How can there be a base, if it is a 2D triangle?(1 vote)
- what if the triangle is equilateral?(1 vote)
- You can't draw an equilateral triangle with all of its vertices on the grid. At least one coordinate must be irrational.(2 votes)

## Video transcript

- [Voiceover] What I would like to do is find the area of this green triangle. And if you get so inspired, and 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 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. I'm gonna 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 looked 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. So that is the base b. And now the height, I guess you could say this, if you were to drop a penny from here, it's
sitting outside the triangle. So it looks different from this one. But this would still be the height. This would still be the height. Right over here, ya do the same thing. One half 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 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 and 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, 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 three. 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. 'Cause once again, we can view this yellow line, or now I have
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 two. One half times two is one times three. This is going to be equal to three. So the area of the whole thing is gonna be this area of three plus this area of six. It's gonna be the area of nine. Area is equal to nine. Now that's one way you could do it. Is you could just break
it up into triangles where you could figure out
the base and the height. Another way, and this is you can kinda view it as a maybe a trickier way. Or ya kinda 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 to show
you the other approach. The other way that we can tackle this. So, the other way we could tackle it. I'm gonna 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 triangle is embedded inside of our rectangle. So let me draw the rectangle. Let me draw a, let me draw
the broader rectangle. And I think you might
see where this is going. 'Cause as soon as you draw that, that bigger rectangle, then you see that that rectangle is made up of the triangle that we're
tryin' 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 gonna be four times six, those little mentions of the rectangle. Four times six. So the area of the entire rectangle is 24. And then you subtract out the area of the purple, the blue
and the yellow rectangles. The purple, the blue and
the yellow triangles, then you're gonna 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 gonna subtract it out. It's gonna be one half, six, the height right over here. If we view this as a height of six. And it's a base, right over here is three. So it's gonna be one half 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 one half. Let's see, you could say height is one. 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 one half. Let's see we could make this base two. One half times two times, and this height is four. Two times four. So what's this gonna be? Well, let's see, one half
times six times three. That's three times three. That's going to be nine. One half times one times four. That's going to be two. And then one half times two times four. Well that's just going to be four. And so we're left with 24 minus
nine minus two minus four. So, this we're gonna do the same color. Minus four, so what is that? 24 minus nine is 15. 15 minus two is 13. 13 minus four is equal to nine. It's equal to nine. So that's the other way, or another way to get the area of this green triangle.