Stacked squares in a triangle

Geometry Level 3

Given a triangle with a base of 9 inches and height of 8 inches,

Suppose we stack squares infinitely in the triangle as shown in the figure above, what is the total area of all the squares in the triangle in square inches ?


The answer is 23.04.

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Vijay Simha by Vijay Simha , 47, USA

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2 solutions

Rab Gani
Jun 3, 2020

Using similarity , and let the sides of squares of decreasing values are a, b, etc. So we find the total height of similar trapezium a/(1-b/a) = 8. And from similarity we have b/a = a/9. From these 2 equations, we find a=72/17. The total area of squares = a^2 /[1 - (b/a)^2] = 23.04

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Vijay Simha
May 27, 2015

Clearly the length of the side of the base (orange) square is (b h/(b+h)), and its area is A1 = (b h/(b+h))^2

The length of the side of the white square is (b h^2/(b+h)^2) and its area is A2 = (b h^2/(b+h)^2)^2

Similarly A3 is (b*h^3/(b+h)^3)^2

A1, A2, A3 .... infinity form a geometric sequence the sum of whose terms is given by a/(1-r) where a = A1 and r is the common ratio which is equal to h^2/(b+h)^2

Total area is therefore bh^2/(b + 2h).

Substituting the values of b = 9 and h = 8,

We get the total Area to be = 9*64/(9+16) = 576/25 = 23.04

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