## Diagonal of a Box

# Suppose we have a
box with length, width, and height given, as indicated.

# We want to know the
length of the diagonal d, shown in red on the diagram.

We can first compute the diagonal of the face of the box whose length l and
width w are given. Let's call this diagonal c.^{ }

c = √ (l^{2} + w^{2})^{ }

Notice that c is the leg of a new right triangle whose other leg is the height
h of the box, and whose diagonal d is the diagonal of the box.^{ }For
this right triangle, we know^{ }

d = √(c^{2} + h^{2})^{ }

But since c = √ (l^{2} + w^{2}),^{ }

c^{2} = l^{2} + w^{2}, so^{ }

d = √( l^{2} + w^{2} + h^{2})

And we have found how to get the length of the diagonal of a box with
dimensions l, w, and h!^{ }