Pythagorean Theorem and Cubes

Date: 02/14/98 at 13:38:38
From: Kelly Shott
Subject: Pythagorean Theorem as it applies to cubes

In a cube if a diagonal is drawn, how long must each side be using the 
Pythagorean Theorem? Simple, right? However in the diagram the 
diagonal is not simply drawn from one corner to the other of one of 
the one dimensional sides of the cube. It is drawn from the front top 
corner to the back bottom corner, making it necessary to take into 
account the length of the diagonal of the bottom panel. How is the 
Pythagorean Theorem used in relation to cubes and not simply squares?

Date: 02/15/98 at 10:31:51
From: Doctor Sam
Subject: Re: Pythagorean Theorem as it applies to cubes

Kelly,

The diagonal you are describing is called a "space diagonal" because 
it moves through the inside space of the cube from one corner to the 
opposite corner.  In order to apply the Pythagorean Theorem we need to 
find a right triangle with that diagonal as one of its sides.

Imagine slicing a cake in the shape of a cube in half. But instead of 
the normal slice, which is parallel to one of the cake's edges, slice 
along the diagonal of the top of the cake. The knife will slide 
vertically down two opposite edges of the cube and divide the cake in 
half.

The exposed area inside the cube is a rectangle. The height of the 
rectangle is the side of the cube (the height of the cake) and the 
width of the rectangle is the diagonal of the top of the cake.

The space diagonal that you are interested in is the diagonal of this 
rectangle. You can find the length of this diagonal using the 
Pythagorean theorem in a right triangle whose two sides are (1) a side 
of the cube and (2) the diagonal of the top face of the cube and whose 
hypotenuse is the space diagonal.

Of course, you don't know the length of the diagonal of the top face, 
but you can find it by using the Pythagorean Theorem too. By applying 
the theorem twice, once in a right triangle lying on the surface of 
the cube and then again in a right triangle in the plane of the 
vertical slice, you can discover that the space diagonal's length is 
the side of the cube times the square root of 3.  

By the way, the "cake" doesn't have to be in the shape of a cube; any 
rectangular cake will do.

I hope that helps.

-Doctor Sam,  The Math Forum
 Check out our Web site  http://mathforum.org/dr.math/