The Hermit Crab problem

Problem

A spiral of consecutive right-angled triangles, each built on the hypotenuse of the previous one, with legs $3, 4, 5$.

1. Prove that the triangles are similar.
2. Find the similarity ratio between two consecutive triangles.
3. Compute the sum of the areas of the infinitely many triangles.

Solution

1. They are similar because they have congruent angles (all right-angled, with the same angle at the centre of the spiral).

2. The longer leg of one triangle becomes the hypotenuse of the next: the similarity ratio is $k = \dfrac{4}{5}$.

3. The ratio between the areas of two consecutive triangles is $k^2 = \dfrac{16}{25}$. The area of the first triangle is $A_0 = \dfrac{3\cdot 4}{2} = 6$.

The areas form a geometric series: $A_{\text{tot}} = 6 + 6\cdot\frac{16}{25} + 6\cdot\left(\frac{16}{25}\right)^2 + \cdots = 6\cdot\sum_{n=0}^{\infty}\left(\frac{16}{25}\right)^n.$

Trick: let $G = 1+\frac{16}{25}+\left(\frac{16}{25}\right)^2+\cdots$ and factor out $\frac{16}{25}$: $G = 1 + \frac{16}{25}\cdot\underbrace{\left(1+\frac{16}{25}+\cdots\right)}_{=\,G} = 1+\frac{16}{25}\,G \implies \frac{9}{25}\,G = 1 \implies G = \frac{25}{9}.$

Therefore $A_{\text{tot}} = 6\cdot\dfrac{25}{9} = \boxed{\dfrac{50}{3}}$.

Links

Topics: Similarity
Concepts: Scaling laws · Similarity ratio · Geometric series · Similar triangles
Methods: Similarity criteria
Skills: Calculating · Proving · Synthetic geometry
Exercise type: Geometric problem