A cube has a space diagonal of \(6\sqrt{3}\) cm. What is the volume of the cube?
["# Finding the Volume of a Cube When Given Its Space Diagonal", "When working with 3D geometry, the relationship between a cube’s dimensions and its space diagonal provides essential insights for solving problems in architecture, engineering, and geometry. In this article, we explore how to calculate the volume of a cube when you know the length of its space diagonal.", "## What Is a Space Diagonal?", "The space diagonal of a cube connects two opposite vertices through the interior of the cube, passing entirely within its structure. Unlike face diagonals that lie on a cube’s flat surfaces, the space diagonal reflects the cube’s full three-dimensional extent.", "For a cube with side length ( s ), the formula for the space diagonal ( d ) is:", "[\nd = s\sqrt{3}\n]", "This formula arises from the Pythagorean theorem applied in three dimensions:\n[\nd = \sqrt{s^2 + s^2 + s^2} = \sqrt{3s^2} = s\sqrt{3}\n]", "## Given: Space Diagonal = (6\sqrt{3}) cm", "We are told the space diagonal of the cube is (6\sqrt{3}) cm. Using the space diagonal formula:", "[\ns\sqrt{3} = 6\sqrt{3}\n]", "To find the side length ( s ), divide both sides by ( \sqrt{3} ):", "[\ns = 6 \ ext{ cm}\n]", "## Calculating the Volume", "The volume ( V ) of a cube is given by:", "[\nV = s^3\n]", "Substituting ( s = 6 ) cm:", "[\nV = 6^3 = 216 \ ext{ cm}^3\n]", "## Conclusion", "When a cube has a space diagonal of (6\sqrt{3}) cm, its side length is 6 cm, and therefore its volume is 216 cubic centimeters. Understanding this geometric relationship helps simplify many real-world applications where spatial measurement is crucial.", "---", "Key Takeaway: Given the space diagonal of a cube formula, simply solve for side length, then compute volume using ( V = s^3 ). For ( d = 6\sqrt{3} ) cm, volume = (216 \ ext{ cm}^3)."]
