A rectangular prism has dimensions 5 cm, 12 cm, and 13 cm. What is its surface area?
["Understanding the Surface Area of a Rectangular Prism: A Practical Example with Dimensions 5 cm × 12 cm × 13 cm", "When dealing with three-dimensional geometric shapes, understanding the surface area is essential for fields such as engineering, packaging design, architecture, and everyday problem-solving. One commonly studied shape is the rectangular prism—also known as a rectangular box—which features six rectangular faces. In this article, we’ll dive into the surface area of a rectangular prism with dimensions 5 cm, 12 cm, and 13 cm, providing a clear, step-by-step solution and practical insight.", "---", "### What Is a Rectangular Prism?", "A rectangular prism is a solid figure with six rectangular faces. All interior angles are right angles (90°), and opposite faces are congruent rectangles. Its key measurements—length, width, and height—directly determine its surface area.", "---", "### The Formula for Surface Area", "The surface area ( S ) of a rectangular prism is calculated using the formula:", "[\nS = 2(lw + lh + wh)\n]", "Where:\n- ( l ) = length (in cm)\n- ( w ) = width (in cm)\n- ( h ) = height (in cm)", "This formula adds the areas of all six faces: two faces of size ( l \ imes w ), two faces of size ( l \ imes h ), and two faces of size ( w \ imes h ).", "---", "### Applying the Formula to Dimensions 5 cm, 12 cm, and 13 cm", "Given:\n- Length (( l )) = 13 cm\n- Width (( w )) = 12 cm\n- Height (( h )) = 5 cm", "Now substitute into the formula:", "[\nS = 2 \left( (13 \ imes 12) + (13 \ imes 5) + (12 \ imes 5) \right)\n]", "Calculate each product:", "- ( 13 \ imes 12 = 156 , \ ext{cm}^2 )\n- ( 13 \ imes 5 = 65 , \ ext{cm}^2 )\n- ( 12 \ imes 5 = 60 , \ ext{cm}^2 )", "Add these areas:", "[\n156 + 65 + 60 = 281 , \ ext{cm}^2\n]", "Now multiply by 2:", "[\nS = 2 \ imes 281 = 562 , \ ext{cm}^2\n]", "---", "### Final Result", "The surface area of a rectangular prism with dimensions 5 cm, 12 cm, and 13 cm is 562 square centimeters.", "---", "### Why Knowing Surface Area Matters", "- Manufacturing & Packaging: Knowing surface area helps estimate material needs for making boxes, labels, or coatings.\n- Thermal and Electrical Applications: Surface area affects heat transfer and conductivity in 3D objects.\n- Geometry Education: Practicing surface area formulas deepens spatial reasoning and mathematical fluency.", "---", "### Conclusion", "Calculating the surface area of a rectangular prism is straightforward with its three dimensions. By applying the formula ( S = 2(lw + lh + wh) ), we efficiently determine that a prism measuring 5 cm × 12 cm × 13 cm has a surface area of 562 cm². Mastering this concept equips learners and professionals with a vital tool for solving real-world geometric challenges.", "---", "Keywords for SEO: rectangular prism surface area, surface area of rectangular prism formula, rectangular prism 5x12x13 cm, math calculation guide, geometry formulas, 3D shape surface area, how to calculate prism surface area", "Optimizing technical content with precise geometric facts and real-world applications enhances search visibility and user value—perfect for students, teachers, and math enthusiasts."]
