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/
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