To find the length of a diagonal of a cube, you can use the formula: *diagonal length = side length √3**.
Here's a breakdown of how to arrive at this formula:
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Consider a cube: Imagine a cube with each side having a length of 's'.
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Face Diagonal: First, find the diagonal of one of the cube's faces. This forms a right triangle with two sides of the cube. Using the Pythagorean theorem (a² + b² = c²), the face diagonal is √(s² + s²) = √(2s²) = s√2.
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Cube Diagonal (Space Diagonal): Now, the diagonal of the cube (also known as the space diagonal) forms another right triangle. One leg of this triangle is the face diagonal (s√2), and the other leg is one of the cube's sides (s). Again, using the Pythagorean theorem:
(s√2)² + s² = diagonal²
2s² + s² = diagonal²
3s² = diagonal²
diagonal = √(3s²) = s√3
Therefore, the length of the diagonal of a cube is the side length multiplied by the square root of 3.
Example:
If the side length of a cube is 5 inches, then the length of its diagonal is 5 * √3 inches, which is approximately 8.66 inches.
