Step by step
The question
Question: find the diagonal of a box 4 by 7 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 corners are square corners.
So its diagonal d and the sides 4 and 7 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 4 squared plus 7 squared.
4 times 4 is 16.
7 times 7 is 49.
Together 65.
65 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 9, 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 9 squared.
d squared is 65.
9 times 9 is 81.
Together 146.
146 is not a square number.
A calculator gives 12.083, and so on.
To the nearest tenth, D is about 12.1.
3 Check
Let's check in one go: d squared was 4² + 7².
So D squared is 4² + 7² + 9².
16 plus 49 plus 81 is 146, the same.
So the answer is right!
Let's look back.
First: floor diagonal.
Then: diagonal of the box.
Last: check.