A rectangular prism has dimensions 4 cm, 5 cm, and 6 cm. What is the length of its space diagonal?
["How to Calculate the Space Diagonal of a Rectangular Prism (Dimensions 4 cm, 5 cm, 6 cm)", "Understanding the dimensions of a rectangular prism is essential in geometry, engineering, and real-world applications like architecture, packaging, and design. If you’ve ever wondered, “What is the length of the space diagonal of a rectangular prism with dimensions 4 cm, 5 cm, and 6 cm?”, this guide provides a clear and step-by-step solution.", "### What Is a Rectangular Prism?", "A rectangular prism is a three-dimensional shape with six rectangular faces. It has three pairs of equal opposite faces and straight edges. Each corner (vertex) connects to three edges measuring 4 cm, 5 cm, and 6 cm in length—exactly its defining dimensions.", "### The Space Diagonal Explained", "The space diagonal (also called the 3D diagonal) is the longest straight line connecting two opposite vertices (corners) through the interior of the prism. Unlike face diagonals—which lie in flat surfaces—the space diagonal stretches through the volume.", "For a rectangular prism with length ( l ), width ( w ), and height ( h ), the space diagonal ( d ) is calculated using the 3D Pythagorean theorem:", "[\nd = \sqrt{l^2 + w^2 + h^2}\n]", "### Applying the Formula", "Given:\n- ( l = 4 , \ ext{cm} )\n- ( w = 5 , \ ext{cm} )\n- ( h = 6 , \ ext{cm} )", "Substitute values into the formula:", "[\nd = \sqrt{4^2 + 5^2 + 6^2} = \sqrt{16 + 25 + 36} = \sqrt{77}\n]", "### Calculating the Exact Value", "[\n\sqrt{77} \approx 8.775 , \ ext{cm}\n]", "### Why Is This Important?", "Knowing the space diagonal helps in:", "- Determining if an object fits inside constrained spaces\n- Calculating internal distances for structural designs\n- Solving real-life problems like packaging dimensions or shipping logistics", "### Final Answer", "The length of the space diagonal of a rectangular prism with dimensions 4 cm × 5 cm × 6 cm is:", "[\n\boxed{\sqrt{77} \approx 8.78 , \ ext{cm}}\n]", "Mastering this calculation strengthens your spatial reasoning and problem-solving skills in geometry—skills that apply far beyond the classroom."]
