Step by step
The question
Question: find the diagonal of a box 4 by 5 by 6.
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 4 and 5.
Pythagoras: d squared is 4 squared plus 5 squared.
4 times 4 is 16.
5 times 5 is 25.
Together 41.
41 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 6, and the diagonal D of the box.
Pythagoras again: D squared is d squared plus 6 squared.
d squared is 41.
6 times 6 is 36.
Together 77.
77 is not a square number.
A calculator gives 8.775, and so on.
To the nearest tenth, D is about 8.8.
3 Check
Let's check: the two steps are the same as l squared plus w squared plus h squared.
16 plus 25 plus 36 is 77, the same D squared.
So the answer is right!
Let's look back.
First: floor diagonal.
Then: diagonal of the box.
Last: check.