Series: Finding the area of infinitely many triangles

Question

Compute the total area of the (infinitely many) triangles in the Figure.

The height of all triangles is $\dfrac67$. $x-$values for the bases of the triangles from left to right are as follows: $\dfrac{27}{64}$, $\dfrac9{16}$, $\dfrac34$, $1$, $\cdots$

Using the formula for the area of a triangle $\left(A = \dfrac12BH\right)$ as well as the fact that all the triangles have the same height of $\dfrac67$, I know that the area is equal to $\dfrac37B$. I'm not quite sure where to go from there when it comes to solving this, however.

Answer 1

Since the base $x$-values are increasing by a factor of $4/3$ in every new triangle, the ratio is greater $1$ and thus the bases are getting larger indefinitely. The area is not finite.

Answer 2

From the picture, it looks as though the triangles are getting smaller and approaching the $y$-axis. IF SO, then you have a geometric series for their area which converges: $$\sum_{n=1}^\infty\frac{1}{2}\frac{6}{7}\left(\frac{3^{n-1}}{4^{n-1}}-\frac{3^n}{4^n}\right)=\sum_{n=1}^\infty\frac{1}{2}\frac{6}{7}\left(\frac{4}{3}-1\right)\frac{3^n}{4^n}=\frac{1}{2}\frac{6}{7}\frac{1}{3}3=\frac{3}{7}$$

Answer 3

Note: OP has left an ambiguity in which direction the infinite triangles go, from left to right or from right to left. I've answered for both the cases


I assume that the triangle's bases are decreasing indefenitely from right (at $x=1$) to $x=0$. Then, the area of the triangle decreases by a factor of $3/4$ from right to left. Assume the total area of triangles is $A$. If you exclude the largest triangle on the right, then the remaining area is just a scales down version of the original picture. Thus, it's area will be $\frac{3}{4}A$. Thus, we can say $$\text{Area of Largest Triangle}+\frac{3}{4}A=A$$ $$3/28=1/4A$$ $$A=\frac{3}{7}$$

If the trangle's bases are increasing indefenitely towards the right, then the area is diverging and we cannot find it.

Answer 4

Interesting question.

The area of a triangle is half base times height, and since all triangles have equal height, the total area is half total base times height, i.e. $$\frac 12\cdot 1\cdot \frac 67 = \color{red}{\frac 37}$$

The geometric series summation is a diversion :)