Inscribing Boxes One After Another

Geometry Level 3

$\huge {\boxed {\boxed {\boxed {\boxed {\boxed {\boxed {\cdots}}}}}}}$

The figure above is made of infinite squares. Every inscribed square's diagonal is Half the diagonal of the circumscribing square. What is the total area of all squares?

Clarification: The figure shown above shows infinite squares incribed inside one another, with the diagonal of each inscribed square being half of the square outside it.

The area of the largest square is 1 unit.

Note : Figure not drawn to scale.

The answer is 1.333.

1 solution

Mehul Arora
Mar 3, 2016

The area of the largest square is $1 \text {unit}$

Each successive square has an area = $\dfrac {1}{4}$ the area of the square inscribing it.

Let the diagonal of the largest square be $x$

Area = $\dfrac {x^2}{2}$

Each successive square swuare has an area of $\dfrac {x^2}{4 \times 2}$

So, that areas of the squares are as follows:

$1+ \dfrac {1}{4} + \dfrac {1}{16}+....$

Which leads to an infinite GP. So, total = $\dfrac {1}{1- \dfrac {1}{4}} = \dfrac {4}{3} = 1.33$

Hey I did it the same way but I assumed that X is 1. Is there like some reason for it because different values will give different answers...... Btw same solution :)

abc xyz - 5 years, 3 months ago

It is given that the area of the largest square is 1 unit :)

Mehul Arora - 5 years, 3 months ago

Oh ok didn't see that..... But managed to solve the problem !!! Thanks

abc xyz - 5 years, 3 months ago

Inscribing Boxes One After Another

The answer is 1.333.

1 solution

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