Find the length of the diagonal of a rectangular prism with dimensions 3 cm, 4 cm, and 12 cm.
["# Finding the Length of the Diagonal of a Rectangular Prism (Dimensions 3 cm, 4 cm, 12 cm)", "When working with 3D geometry, one common problem is calculating the diagonal length of a rectangular prism. Whether you're designing packaging, engineering furniture, or solving math problems, knowing how to find the diagonal efficiently is essential. In this article, we’ll explore how to calculate the diagonal of a rectangular prism with dimensions 3 cm, 4 cm, and 12 cm using a straightforward geometric formula.", "## What Is the Diagonal of a Rectangular Prism?", "The diagonal of a rectangular prism connects two opposite corners and passes through the interior of the solid. It is the longest line segment that fits entirely within the prism. For a prism with length ( l ), width ( w ), and height ( h ), the diagonal ( d ) can be found using the 3D Pythagorean theorem:", "[\nd = \sqrt{l^2 + w^2 + h^2}\n]", "## Step-by-Step Calculation", "Given the dimensions:", "- Length ( l = 12 ) cm\n- Width ( w = 4 ) cm\n- Height ( h = 3 ) cm", "We substitute into the formula:", "[\nd = \sqrt{12^2 + 4^2 + 3^2}\n]", "Compute each square:", "[\n12^2 = 144,\quad 4^2 = 16,\quad 3^2 = 9\n]", "Add the squares:", "[\n144 + 16 + 9 = 169\n]", "Now take the square root:", "[\nd = \sqrt{169} = 13\n]", "## Result", "The length of the diagonal of the rectangular prism is 13 cm.", "## Why This Formula Works", "This method applies because the diagonal forms the hypotenuse of a right triangle whose legs are the face diagonal and the height. Squaring each dimension ensures all spatial distances are accounted for unambiguously in three-dimensional space.", "## Practical Applications", "- Packaging design: To ensure labels or protective covers fit fully.\n- Architecture and construction: For accurate measurements inside cubic spaces.\n- Mathematics and physics: Solving problems involving spatial relationships.", "## Conclusion", "Calculating the diagonal of a rectangular prism is simple when using the formula ( d = \sqrt{l^2 + w^2 + h^2} ). For a prism measuring 3 cm × 4 cm × 12 cm, the diagonal measures exactly 13 centimeters — a key measurement useful in both academic and real-world applications.", "If you frequently work with prisms or spatial geometry, memorizing this shortcut will save time and reduce errors. Whether you’re a student, engineer, or DIY enthusiast, this diagonal formula is an essential tool in your geometric toolkit."]
