Step by step
The question
Question: find the diagonal of a box 5 m by 6 m by 8 m.
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 5 and 6.
Pythagoras: d squared is 5 squared plus 6 squared.
5 times 5 is 25.
6 times 6 is 36.
Together 61.
61 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 8, and the diagonal D of the box.
Pythagoras again: D squared is d squared plus 8 squared.
d squared is 61.
8 times 8 is 64.
Together 125.
125 is not a square number.
A calculator gives 11.180, and so on.
To the nearest tenth, D is about 11.2 m.
3 Check
Let's check: the two steps are the same as l squared plus w squared plus h squared.
25 plus 36 plus 64 is 125, the same D squared.
So the answer is right!
Let's look back.
First: floor diagonal.
Then: diagonal of the box.
Last: check.