To find the length of the main diagonal (space diagonal) of a cube, you can use a simple formula derived from the Pythagorean theorem. The exact answer for the length of the space diagonal of a cube with side length 's' is s times the square root of 3 (s√3).
Understanding Cube Diagonals
A cube has different types of diagonals:
- Face Diagonal: A diagonal across one of the square faces of the cube.
- Space Diagonal: A diagonal that passes through the interior of the cube, connecting opposite vertices. This is typically what people mean when asking for "the diagonal of a cube".
Formula for the Space Diagonal
The length of the space diagonal (d) of a cube can be calculated using the length of its side (s).
The formula is:
d = s√3
Where:
dis the length of the space diagonalsis the length of one side (edge) of the cube√3is the square root of 3 (approximately 1.732)
How the Formula is Derived
The formula s√3 is found by applying the Pythagorean theorem twice:
-
Find the face diagonal: Consider one face of the cube. It's a square with side 's'. The diagonal of this face (
d_face) forms a right triangle with two sides of the square. Using the Pythagorean theorem (a² + b² = c²):
d_face² = s² + s²
d_face² = 2s²
d_face = √ mistakes(2s²)
d_face = s√2 -
Find the space diagonal: Now, consider a right triangle formed by:
- The face diagonal (
d_face = s√2) of the bottom face. - A vertical side of the cube (
s). - The space diagonal (
d).
These three lengths form a right triangle where the space diagonal is the hypotenuse. Applying the Pythagorean theorem again:
d² = (d_face)² + s²
d² = (s√2)² + s²
d² = (s² * 2) + s²
d² = 2s² + s²
d² = 3s²
d = √3s²
d = s√3
- The face diagonal (
This derivation shows how the formula s√3 is obtained.
Example Calculation
Let's say you have a cube with a side length of 5 cm.
To find the space diagonal:
s = 5 cm
d = s√3
d = 5 cm * √3
d ≈ 5 cm * 1.732
d ≈ 8.66 cm
So, the space diagonal of a cube with a side length of 5 cm is exactly 5√3 cm or approximately 8.66 cm.
The Exact Answer
As mentioned in the provided video reference, obtaining the length using the formula s√3 gives the exact answer for the diagonal length of a cube. The steps involving the Pythagorean theorem lead directly to this precise mathematical expression.
| Cube Dimension | Formula | Example (s=5) |
|---|---|---|
| Side Length | s |
5 cm |
| Face Diagonal | s√2 |
5√2 cm |
| Space Diagonal | s√3 |
5√3 cm |
Finding the diagonal of a cube simplifies to knowing the side length and multiplying it by the square root of 3.
