Longest pole that fits in a box 9 by 12 by 8

Step by step

The question

Question: what is the longest pole that fits in a box 9 by 12 by 8?

The longest thing that fits goes from a bottom corner to the top corner on the other side: the diagonal through the box.

1 Floor diagonal

First the floor.

Its diagonal d makes a right triangle with the sides 9 and 12.

Pythagoras: d squared is 9 squared plus 12 squared.

9 times 9 is 81.

12 times 12 is 144.

Together 225.

15 times 15 is 225.

So d is 15.

Careful!

We cannot just add the sides.

A diagonal is a short cut: Pythagoras, twice.

2 Diagonal of the box

Now a new right triangle stands up: the floor diagonal d, the height 8, and the diagonal D of the box.

Pythagoras again: D squared is d squared plus 8 squared.

d squared is 225.

8 times 8 is 64.

Together 289.

17 times 17 is 289.

So D = 17.

3 Check

Let's check: the two steps are the same as l squared plus w squared plus h squared.

81 plus 144 plus 64 is 289, the same D squared.

So the answer is right!

Let's look back.

First: floor diagonal.

Then: diagonal of the box.

Last: check.