Step by step
The question
Question: what is the longest pole that fits in a box 4 by 4 by 7?
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 4 and 4.
Pythagoras: d squared is 4 squared plus 4 squared.
4 times 4 is 16.
4 times 4 is 16.
Together 32.
32 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 7, and the diagonal D of the box.
Pythagoras again: D squared is d squared plus 7 squared.
d squared is 32.
7 times 7 is 49.
Together 81.
9 times 9 is 81.
So D = 9.
3 Check
Let's check: the two steps are the same as l squared plus w squared plus h squared.
16 plus 16 plus 49 is 81, the same D squared.
So the answer is right!
Let's look back.
First: floor diagonal.
Then: diagonal of the box.
Last: check.