# For Geeks & Brainiacs #09- Tightly Packed -

There are three gift boxes: a small cube-shaped box, a sphere-shaped box, and a large cube-shaped box.
The small cube-shaped box has a volume of

3,375 cm³. If the

*small* cube-shaped
box is placed inside the sphere-shaped box, it's a perfect fit. And if the sphere-shaped box is placed
inside the

*large* cube-shaped box, it's also a perfect fit.

Ignoring the thickness of the walls of the boxes, can you find the volume of the large cube-shaped box?

**HINT**: You will need to use Pythagoras' theorem to help you find the length of the longest side
(c) of a triangle with a right angle (90°):

**a²** +

**b²** =

**c²**
**HINT 2**: To find the diameter of the sphere-shaped box, you must find the distance that separates
the two opposing corners of the small cubical box (i.e. the

*space diagonal* or

*triagonal*).

**ANSWER**:

*17,535.47 cm³.*
**EXPLANATION**: Because the small cube-shaped box is a perfect fit inside the sphere-shaped box, its
corners will touch the inside of the sphere-shaped box. Therefore, the diagonal of the small cube-shaped
box from one corner to its opposing corner (termed the

*space diagonal* or

*triagonal*) will
correspond to the diameter of the sphere-shaped box. This is represented in the image below by line y,
which is the distance between opposing corners A and B. Since the small cube-shaped box has a volume of
3,375 cm³, each side has a length of 15 cm (cube root of 3,375).

Using Pythagoras' theorem, we can determine the length of

**x**.

**15² + 15² = x²**
**x = 21.21 cm**
Now that we have

**x**, we can determine

**y**.

**15² + 21.21² = y²**
**y = 25.98 cm** (the diameter of the sphere-shaped box).

Since the sphere-shaped box is a perfect fit inside the large cube-shaped box, its diameter represents the
length of each side of the large cube-shaped box. The volume of the large cube-shaped box is therefore:

**25.98³ = 17,535.47 cm³**
Do you have a

suggestion for this puzzle (e.g. something that should
be mentioned/clarified in the question or solution, bug, typo, etc.)?