Diagonal of a box 12 by 16 by 21

Step by step

The question

Question: find the diagonal of a box 12 by 16 by 21.

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 12 and 16.

Pythagoras: d squared is 12 squared plus 16 squared.

12 times 12 is 144.

16 times 16 is 256.

Together 400.

20 times 20 is 400.

So d is 20.

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 21, and the diagonal D of the box.

Pythagoras again: D squared is d squared plus 21 squared.

d squared is 400.

21 times 21 is 441.

Together 841.

29 times 29 is 841.

So D = 29.

3 Check

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

144 plus 256 plus 441 is 841, the same D squared.

So the answer is right!

Let's look back.

First: floor diagonal.

Then: diagonal of the box.

Last: check.