Find the Total Area of This Triangle! | Math Puzzle

Added: 2026-05-11

[00:00:05]We will start our problem with a triangle. Then  we will pull a line that divides this triangle.

[00:00:13]It will divide it in such a way that on the top  we will have another triangle and on the bottom we will have a quadrilateral, and we will pull  two diagonals of this quadrilateral. In this way, we will divide our big triangle into five smaller  separate triangles, as you can see on the sketch.

[00:00:35]Now, if the area of the top triangle equals 3, the  area of the triangle beneath it equals 1, and the area of the triangle at the bottom equals 4. The  question is, can we somehow calculate the area of the entire triangle? So, if you guys want to try  this problem out for yourself, you can pause the video right now and then come back and check the  solution as always. So, let's solve this one. I'm not going to label the vertices of this triangle  because I think that we can get away without it.

[00:01:14]And I'm going to use two different, let's call  them, geometrical theorems to solve this problem.

[00:01:21]The first one will include a triangle. And let's  say that if we divide this triangle into two separate smaller triangles, we can label the area  of the triangle on the left as A1 and the area of the triangle on the right as A2. And let's say  that the base of the triangle on the left is a and the height of this triangle is h. We can calculate  the area of the triangle A1 as half of the product of the base a and the height h. Now we can rewrite  this a little bit as A1 / a, which equals h / 2. Now, let's take a look at the triangle on the  right and let's label the base of this triangle as B. Now we can use the same formula, but this time  it will be b * h over 2. And you can notice that both of those triangles share the same height or  the same altitude, if you will. So let's rewrite the second formula in the same manner as the  first one and get the following. Now you can notice that both of those formulas are equal h /  2. So we can say that both of them are equal to each other. And we can rewrite this a little bit  to get that A1 / A2 equals a / b. And this is our first super important formula, which we are going  to use on our sketch. So this formula basically states that the relation between the areas of  any two triangles that share the same altitude equals the relation between their bases. So let's  find some triangles with the same altitude on our sketch. But before that, let's label the remaining  two areas of our unknown triangles as X and Y.

[00:03:25]Now let's take a look at the triangle on the top  with the area three. This light green triangle.

[00:03:32]And let's also take a look at the triangle which  consists of triangles with the area 1 and Y. As you can see, those two triangles share the same  altitude. So let's label the base of the triangle at the top as a and the base of the triangle at  the bottom as b. Now we can use our formula. We can say that the relation between the areas  of those two triangles equals a over b or the relation between their bases. Now, let's find  another two triangles, but let's keep our bases.

[00:04:11]So let's try to find two other triangles that  share the same height but also have the same bases of a and b. And you can notice that we have  this big triangle with the combined area of 3, 1, and X, and the triangle at the bottom with the  combined area of y and four. Those two triangles again share the same altitude. So we can use our  formula on those two as well. We can write this as 3 + 1 + X over 4 + Y, which equals a / b. Now, if  you take a look at our equations, you can notice that both equations are equal a / b. So we can  solve this as a system of simultaneous equations.

[00:05:00]We can say that those two equations are then equal  to each other. Then we can cross-multiply this, and we can simplify our rightmost term to get  the following. Then we can expand our equation by multiplication and put the values of x and y on  the left side, and work through this. And finally, we can multiply everything by -1 to get rid of the  minus sign. And we will get that X + Y = 8 - XY., But you can notice that we do not have the value  of XY. So we're going to use another theorem, and I will call this one the intersecting  diagonals area theorem in a quadrilateral.

[00:05:51]As you noticed, I said that I will call this  theorem this way because, to be honest with you, I have no idea how this theorem is actually  called. I don't even know if it has a name. So if you guys know the name of this theorem, please  write it down in the comment section below. But I use the most descriptive way that I could think  of to name this theorem. So this theorem basically applies to any quadrilateral, and it states that  if the diagonals of any quadrilateral intersect, they divide into four smaller triangles, and  the areas of those triangles satisfy that s1 * s3 = to s2 * s4. So basically, the product  of the opposite areas is always constant, and we can use this in our case. If you take a look at  our sketch, you can see that we have this really nice quadrilateral, and we have two intersecting  diagonals inside it. So if we use this theorem, we can conclude that X * Y = 1 * 4 since those are  the products of the opposite areas. Now, from here we can conclude that XY = 4, and we can substitute  this value into our first equation. In this way, we will get the following, and X + Y will be  equal to 4. Now, this is great because remember, we're looking for the area of the entire red  triangle. And this area equals the sum of all individual areas of the small triangles inside.  And we can substitute the value of X + Y with the value of four and get the following. And from  here, we can conclude that the area of our entire red triangle equals 12 square units. And this is  our solution. Well, I really hope you guys enjoyed this problem. If you did, do not hesitate to  leave a like or subscribe to the channel for more content like this. And of course, if you found  another way to solve this problem, please write it down in the comment section as you always do.  And until next time, see you guys and take care.