In a rectangular prism, a **space diagona**l is a line that goes from a vertex of the prism, through the center of the prism to the opposite vertex. That line is also called triagonal or volume diagonal.

A rectangular prism has 4 space diagonals.

The

Pythagorean theorem is useful when we need to find the length of a

**space diagonal** in a rectangular prism. Let's see the following picture:

c^{2}=a^{2}+b^{2}
Find d.

In order to find the value of d, we can use the following right triangle:

In order to find the value of c, we can use this right triangle:

Using the Pythagoran Theorem:

c^{2} = 8^{2}+6^{2}

c^{2} = 100

c = 10

Going back to the first right triangle:

d^{2}=10^{2}+c^{2}

d^{2}=100+100

d^{2}=200

d = in

What is the longest object that will fit inside a 8-by-10-by-12
inch box?

The longest object that will fit inside this box is a segment with the same length as the space diagonal. So we must find the length of the space diagonal of this box.

In order to find the value of d, we can use the following right triangle:

d^{2}=12^{2}+c^{2}

In order to find the value of c, we can use this right triangle:

Using the Pythagoran Theorem:

c^{2}=10^{2}+8^{2}

c^{2}=164

Going back to the first right triangle:

d^{2}=12^{2}+c^{2}

d^{2}=144+164

d^{2}=308

d=17.6 in

The longest object that will fit inside this box is a segment 17.6 in. long