2 Squares, 1 Triangle – What’s The Area?

Added: 2026-05-17

[00:00:00]Hello my lovelies. It's Susanna and today I want to show you how to solve this problem.

[00:00:06]What is the area of the red triangle if the areas of the gray squares are given as 9 square meters and 16 square m. We have our red triangle here. We have gray squares here and here. We are given a right angle here and another one here.

[00:00:30]And we have this line that intersects this line. And this is all the information we have to find the area of the red triangle.

[00:00:42]In general, to find the area of a triangle, there is a formula we can use.

[00:00:48]It is base time height / 2. So we only have to find the base and the height of the triangle to find the area in the end. As a base, I can pick each side of my triangle. So let's just choose this one here as our base.

[00:01:13]And how can I find the height? Then it depends on how your triangle looks like where your height is going to be. So for example, if your triangle looks like this for example and we have our base here, then you always find the heights by going to the vertex opposite the base. So go to this vertex here and draw a line that is perpendicular to my base and this is my height. And in this case my height lies inside my triangle.

[00:01:55]That is different for our triangle but we find the height by taking the exact same steps. So we have our base. We go to the vertex that is opposite my base.

[00:02:12]And now I draw a line down here that is perpendicular to my base. And this is my height of my triangle. And in this case the height lies outside of my triangle.

[00:02:29]But both cases are possible.

[00:02:32]Okay. So now we have B here, H here and we need to find B and H to find the area in the end. So let's see how we can find B and H.

[00:02:45]We haven't used our gray squares yet.

[00:02:49]Let's uh take a look at this one here first. If the area of my square is nine and the side of a square has the same length here and here, then the length of my side has to be three. So that I have 3 * 3, which is 9 for my area. And the same thing here, if my area is going to be 16 of my square, then my sides have to be of length four. So that I have 4 * 4 that equals 16.

[00:03:26]Okay. So my B is part of this triangle and my H is part of this triangle.

[00:03:36]And if we take a look at these two triangles, we have a right angle here and a right angle here, which means that we have two right triangles.

[00:03:48]And what about the other angles?

[00:03:50]This angle here, for example, is actually the same as this angle here.

[00:03:58]Why? Because this line intersects this line. So every time you have a line that intersects another line and you have let's say alpha here then here on the other side you always have alpha as well. These are so-called vertical angles. Which means in our case that if we have alpha here, we also have alpha here.

[00:04:32]Which means now that we have alpha and a right angle here and alpha and a right angle here. So two angles are the same which also tells us that the third angle is the same in both triangles. So if we have beta here, we also have beta here.

[00:04:56]And because we have the same angles in both triangles, these triangles are similar to each other, which is great because then we can use the following.

[00:05:10]If we create the ratio of two sides in one triangle, then this ratio is the same as the ratio of two sides of the other triangle.

[00:05:27]So we go to our first triangle and let's choose this side here, our h. This is the side from beta to the right angle.

[00:05:38]If we look for the corresponding side in this triangle, we also have to go from beta to the right angle. So this here is the corresponding side.

[00:05:51]And if we choose another side, let's take this one here, the three from the right angle to alpha, then the corresponding side here is from the right angle to alpha. So it is our B.

[00:06:07]Let's create the ratios. We start with the green sides. So we take h here and we divide it by the orange side by the three. And the same now in the other triangle. We start with a green side with the four and divide it by the orange side by the b. And now we have an equation. We have h in here. We have b in here. two variables in just one equation is not perfect, but uh maybe we still use it. So maybe let's bring both variables to one side. So let's bring the b here to the other side by multiplying by b here and by b here. So here we have b * h and we divide this by three and on the other side this cancels out and only four is left. And now we want to bring the three to the other side by multiplying by three here and by three here as well. So that this cancels out and only my two variables are left here.

[00:07:25]And here I have 4 * 3 which equals 12. b * h = 12. Let's take this. Let's go back to the beginning where we had our formula here. We now know that b * h= 12. Well, that's actually enough. We only need b * h in here. So, let's replace this by 12. So the area of my triangle is then B * H which is 12. I divide this by 2. So my area is 12 over 2 = 6. So I don't need to know B or H individually. I can just use this product and I'm done. If you like my video, please give it a thumbs up. It helps me a lot. I wish you a wonderful day and I hope to see you in one of our next videos.