Problem Solving

The main diagonal of a cube is the one that cuts through the center of the cube; the diagonal of a face of a cube is not the main diagonal. The main diagonal of any cube can be found my multiplying the length of one side by the square root of 3.
Isn't the formula for this d^2=l^2+w^2+h^2 - or this formula can just be used for rectangles and is not applicable to cubes?

ashblog02 wrote:The main diagonal of a cube is the one that cuts through the center of the cube; the diagonal of a face of a cube is not the main diagonal. The main diagonal of any cube can be found my multiplying the length of one side by the square root of 3.
Isn't the formula for this d^2=l^2+w^2+h^2 - or this formula can just be used for rectangles and is not applicable to cubes?

not rectangles,its cuboid.
Yes same formula can be used, as all the sides in a cube are equal say 'a',
then d^2 = a^2+a^2+a^2 = 3.a^2
Take square root ob both sides, d = sqrt(3). a

Hey guys,

A quick strategic tip on rules like this - if you have them memorized and can apply them quickly you're certainly save a little time, but I think it's must more useful to make sure that you understand why these relationships hold. That way:

1) It's easier to remember because it's not just a list of variables and numbers
2) You can always prove it to yourself if you blank on the formula

This one to me has always been much easier to understand than to memorize. Think of what that longest diagonal will look like:

It's the distance between the lower, front, left corner and the upper, back, right corner (or any other similar distance).

If we look at that as a triangle, the height is going to be the height of the box, from the floor to the ceiling. But the base of that triangle is going to cut through two dimensions - the length and the width. Ultimately, that base is the diagonal of the "floor". So there are two steps to this one:

1) Calculate the diagonal of the floor, which will then be the base of the 3-dimensional triangle that splits the box.

2) Using that as the base and the height of the box as the height, calculate the diagonal of that right triangle to find the longest distance (the hypotenuse of that 3-dimensional triangle).


If you're in a rectangular room, visualize that diagonal and you'll see what I mean. You could very well memorize that rule, but for me I'm much more likely to remember the visual of what those distances represent, and then I can just apply Pythagorean Theorem (which is hopefully something we all have memorized and down cold!) twice.

Brian@VeritasPrep wrote:Hey guys,

A quick strategic tip on rules like this - if you have them memorized and can apply them quickly you're certainly save a little time, but I think it's must more useful to make sure that you understand why these relationships hold. That way:

1) It's easier to remember because it's not just a list of variables and numbers
2) You can always prove it to yourself if you blank on the formula

This one to me has always been much easier to understand than to memorize. Think of what that longest diagonal will look like:

It's the distance between the lower, front, left corner and the upper, back, right corner (or any other similar distance).

If we look at that as a triangle, the height is going to be the height of the box, from the floor to the ceiling. But the base of that triangle is going to cut through two dimensions - the length and the width. Ultimately, that base is the diagonal of the "floor". So there are two steps to this one:

1) Calculate the diagonal of the floor, which will then be the base of the 3-dimensional triangle that splits the box.

2) Using that as the base and the height of the box as the height, calculate the diagonal of that right triangle to find the longest distance (the hypotenuse of that 3-dimensional triangle).


If you're in a rectangular room, visualize that diagonal and you'll see what I mean. You could very well memorize that rule, but for me I'm much more likely to remember the visual of what those distances represent, and then I can just apply Pythagorean Theorem (which is hopefully something we all have memorized and down cold!) twice.

great