How to Find the Longest Internal Diagonal of a Cube

Step-by-Step Guide

1. 1

   Step 1: Open a new Excel workbook and worksheet and draw a unit-cube using the Media Browser "Shapes" tool option -- that means the sides must be equal to 1 unit; that is side s = 1 unit.

   Review:
   The six square shaped exterior surfaces (faces) are equal in dimensions, size, area and have the same shape.
   
   Therefore all faces are congruent.
2. 2

   Step 2: Label 3 consecutive corners (vertices) of the bottom face (the base) as A

   See the figure: label as point D the corner (vertex) above C, at the top of the cube.
   
   The segment CD is at a right angle (90 degrees) to the base. , Let 2 =
   2.
   
   Simplify that; you will find the length of the diagonal of the base, AC.
   
   We have AC = sqrt(2). , Use AC = sqrt(2) and knowing that CD = 1, we substitute these known values into the Pythagorean formula and have the following equation: 2 + 12 = 2 Then let 2 + 12 = 2 + 1 = 3, then 2 =
   2.
   
   Then realize that, the length of the internal diagonal from bottom to top and between opposing corners equals sqrt(3), because 2 = 3 (square root of the squared number) is just that number; let's call the number a, such as 2 = a ) and lengths are always positive numbers. , You also can modify the earlier formula to 2 + 2 =
   2. 2 + 2 = 2, to convert from the unit cube with sides equaling 1, into a multiple of the sides of right triangle ABC with two legs = s*1, and its hypotenuse = s*sqrt(2).
   
   In both cases, the absolute value of s (your cube's side length) is used as the multiplier. ,
3. 3

   Step 3: B and C

4. 4

   Step 4: thus forming triangle ABC.

5. 5

   Step 5: Use the Pythagorean theorem: a2 + b2 = c2

6. 6

   Step 6: for the right triangle ABC where: ` Let 2 + 2 = 2 Then let = 2 + 2 = 1 + 1 = 2

7. 7

   Step 7: for the "left hand side" (LHS) = 2 thus: Examine the length of the RHS = AC squared: 2 = 2.

8. 8

   Step 8: Find the length of the long internal diagonal by using the Pythagorean theorem for right triangle ACD: 2 + 2 = 2

9. 9

   Step 9: where AD is the long internal diagonal we seek.

10. 10

    Step 10: Find the internal diagonal of a cube with a different side length: modify the formula to side s equaling a different number

11. 11

    Step 11: as not for the unit cube but any length of side s; so that each side of the triangle is a multiple of the parts of the unit cube: Let 2 + 2 = 2

12. 12

    Step 12: by multiplication for sides of rt triangle ACD

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    Step 13: and 2 + 2 = 2

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    Step 14: by substitution.

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    Step 15: You're finished!

Detailed Guide

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