Diagonal of a box 3 by 4 by 12

Step by step

The question

Question: find the diagonal of a box 3 by 4 by 12.

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 3 and 4.

Pythagoras: d squared is 3 squared plus 4 squared.

3 times 3 is 9.

4 times 4 is 16.

Together 25.

5 times 5 is 25.

So d is 5.

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 12, and the diagonal D of the box.

Pythagoras again: D squared is d squared plus 12 squared.

d squared is 25.

12 times 12 is 144.

Together 169.

13 times 13 is 169.

So D = 13.

3 Check

Let's check: the two steps are the same as l squared plus w squared plus h squared.

9 plus 16 plus 144 is 169, the same D squared.

So the answer is right!

Let's look back.

First: floor diagonal.

Then: diagonal of the box.

Last: check.