Each side of a cube is 5 inches long, how do you find the lengths of a diagonal of the cube?

1 Answer
Jun 5, 2017

Answer:

The diagonal length of the cube is:
#5sqrt3 " inches"# or #~~8.660" inches"#

Explanation:

To find the diagonal of the cube, there are #2# methods, using a formula or by using the Pythagorus Theorem. So that this explanation is more interesting - and so that you're not just putting numbers into a formula - I am going to tell you how to do the Pythagorus theorem way.

To do this, we need to find the length of the diagonal of the face, in this diagram, from #A# to #B# and then we can construct a triangle with #AB# and #BC# as the two legs, and #CA# as the hypotenuse.
We can find the diagonal by using the Pythagoras Theorem.

First, we need to find the length of #AB#.

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#a^2 + b^2 = c^2#

#s^2 + s^2 = d^2#

#s^2 2 = d^2#

#s = 5#

#5^2 2 = d^2#

We can now use algebra to find out the length of the diagonal of the square, which is the shorter diagonal of the cube.

#5^2 2 = d^2#

#25 xx 2 = AB^2#

#50 = AB^2#

#AB^2 = 50#

#AB = sqrt50#

#sqrt50#

#sqrt50 = sqrt2 xx sqrt(5^2#

#sqrt50 = sqrt2 xx 5#

#sqrt50 = 5sqrt2#

#color(lime)(AB = 5sqrt2#

Now we can construct the triangle and find the overall longer diagonal of the cube.

#a^2 + b^2 = c^2#

#AB^2 + BC^2 = CA^2#

#AB = 5sqrt2#

#BC = 5# because #BC# is simply an edge of the cube.

#(5sqrt2)^2 + 5^2 = CA^2#

Now we can use algebra to find #CA#

#sqrt50^2 + 5^2 = CA^2#

The square root and square of #50# cancel each other out.

#50 + 5^2 = CA^2#

#50 + 25 = CA^2#

#75 = CA^2#

#sqrt75 = sqrt(CA^2#

#sqrt75 = CA#

#sqrt75 ~~ 8.660#

#75 = 3 xx 5^2#

#sqrt75 = sqrt3 xx sqrt(5^2#

#sqrt75 = sqrt3 xx 5#

#sqrt75 = 5sqrt3#

#color(blue)(CA = sqrt75#

#color(blue)(CA = 5sqrt3#

#color(blue)(CA ~~8.660#


And if you wanted to do it the other way, the formula is:

#d = sqrt3 xx a#

#d = sqrt3 xx 5#

#color(blue)(d = 5sqrt3#

Hope this helped. :)