Step by step
The question
Question: find the diagonal of a cube with side 10 cm.
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 10 and 10.
Pythagoras: d squared is 10 squared plus 10 squared.
10 times 10 is 100.
10 times 10 is 100.
Together 200.
200 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 10, and the diagonal D of the box.
Pythagoras again: D squared is d squared plus 10 squared.
d squared is 200.
10 times 10 is 100.
Together 300.
300 is not a square number.
A calculator gives 17.321, and so on.
To the nearest tenth, D is about 17.3 cm.
3 Check
Let's check: the two steps are the same as l squared plus w squared plus h squared.
100 plus 100 plus 100 is 300, the same D squared.
So the answer is right!
Let's look back.
First: floor diagonal.
Then: diagonal of the box.
Last: check.