Longest stick that fits in a box 2 by 10 by 11

Step by step

The question

Question: what is the longest stick that fits in a box 2 by 10 by 11?

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 2 and 10.

Pythagoras: d squared is 2 squared plus 10 squared.

2 times 2 is 4.

10 times 10 is 100.

Together 104.

104 is not a square number.

We do not need d itself: the next step uses d squared.

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 11, and the diagonal D of the box.

Pythagoras again: D squared is d squared plus 11 squared.

d squared is 104.

11 times 11 is 121.

Together 225.

15 times 15 is 225.

So D = 15.

3 Check

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

4 plus 100 plus 121 is 225, the same D squared.

So the answer is right!

Let's look back.

First: floor diagonal.

Then: diagonal of the box.

Last: check.