Diagonal of a box 2 by 3 by 6

Step by step

The question

Question: find the diagonal of a box 2 by 3 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 diagonal d makes a right triangle with the sides 2 and 3.

Pythagoras: d squared is 2 squared plus 3 squared.

2 times 2 is 4.

3 times 3 is 9.

Together 13.

13 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 6, and the diagonal D of the box.

Pythagoras again: D squared is d squared plus 6 squared.

d squared is 13.

6 times 6 is 36.

Together 49.

7 times 7 is 49.

So D = 7.

3 Check

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

4 plus 9 plus 36 is 49, the same D squared.

So the answer is right!

Let's look back.

First: floor diagonal.

Then: diagonal of the box.

Last: check.