Step by step
The question
Question: find the diagonal of a box 5 by 8 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 corners are square corners.
So its diagonal d and the sides 5 and 8 make a right triangle.
Pythagoras: in a right triangle, square the two sides at the square corner and add them.
That makes the long side squared.
Here the long side is the diagonal d.
So d squared is 5 squared plus 8 squared.
5 times 5 is 25.
8 times 8 is 64.
Together 89.
89 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 triangle stands up inside the box: the floor diagonal d, the height 6, and the box diagonal D.
The height goes straight up from the floor, so this corner is square too.
Pythagoras again: D squared is d squared plus 6 squared.
d squared is 89.
6 times 6 is 36.
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.
3 Check
Let's check in one go: d squared was 5² + 8².
So D squared is 5² + 8² + 6².
25 plus 64 plus 36 is 125, the same.
So the answer is right!
Let's look back.
First: floor diagonal.
Then: diagonal of the box.
Last: check.