Step by step
The question
Question: find the diagonal of a box 2 by 6 by 9.
The diagonal goes from a bottom corner to the top corner on the other side, right through the box.
1 Floor diagonal
First the floor.
Its diagonal d makes a right triangle with the sides 2 and 6.
Pythagoras: d squared is 2 squared plus 6 squared.
2 times 2 is 4.
6 times 6 is 36.
Together 40.
40 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 9, and the diagonal D of the box.
Pythagoras again: D squared is d squared plus 9 squared.
d squared is 40.
9 times 9 is 81.
Together 121.
11 times 11 is 121.
So D = 11.
3 Check
Let's check: the two steps are the same as l squared plus w squared plus h squared.
4 plus 36 plus 81 is 121, the same D squared.
So the answer is right!
Let's look back.
First: floor diagonal.
Then: diagonal of the box.
Last: check.