A rectangular prism has dimensions 4 cm by 5 cm by 6 cm. If the prism is painted on all its surfaces, calculate the total surface area.
["How to Calculate the Total Surface Area of a Rectangular Prism (Dimensions 4 cm × 5 cm × 6 cm)", "When working with geometric shapes, understanding surface area is essential—especially for objects like the rectangular prism, commonly used in packaging, architecture, and design. In this article, we’ll explore how to calculate the total surface area of a rectangular prism with dimensions 4 cm, 5 cm, and 6 cm, step by step, ensuring you master this fundamental calculation.", "---", "### What Is a Rectangular Prism?", "A rectangular prism is a three-dimensional shape with six rectangular faces, where each pair of opposite faces are identical. It’s characterized by three dimensions: length, width, and height. For our example, we define:", "- Length ( l = 6 ) cm\n- Width ( w = 5 ) cm\n- Height ( h = 4 ) cm", "---", "### Understanding Surface Area", "Unlike a cube, where all faces are identical, a rectangular prism has three distinct rectangular face types, depending on which dimensions form each pair. Because each pair of opposite faces is congruent, we calculate the area of each unique face and multiply by two.", "The formula for the total surface area ( SA ) of a rectangular prism is:", "[\nSA = 2(lw + lh + wh)\n]", "This formula adds the areas of the three pairs of identical faces:", "- Two faces of dimension ( l \ imes w )\n- Two faces of dimension ( l \ imes h )\n- Two faces of dimension ( w \ imes h )", "---", "### Step-by-Step Calculation", "Let’s plug in the values:", "1. Calculate each pair’s area:", "[\nlw = 6 \ imes 5 = 30\ \ ext{cm}^2\n]\n[\nlh = 6 \ imes 4 = 24\ \ ext{cm}^2\n]\n[\nwh = 5 \ imes 4 = 20\ \ ext{cm}^2\n]", "2. Sum the areas:", "[\nlw + lh + wh = 30 + 24 + 20 = 74\ \ ext{cm}^2\n]", "3. Multiply by 2 for both sides:", "[\nSA = 2 \ imes 74 = 148\ \ ext{cm}^2\n]", "---", "### Final Answer", "The total surface area of a rectangular prism with dimensions 4 cm × 5 cm × 6 cm is 148 square centimeters.", "---", "### Key Takeaways", "- Always identify length, width, and height correctly to apply the surface area formula.\n- A rectangular prism has six rectangular faces in three pairs—no need to compute each face separately.\n- Use the formula ( SA = 2(lw + lh + wh) ) to quickly calculate total surface area.\n- Accurate surface area calculations are vital in fields like construction, shipping, and material estimation.", "Understanding surface area through clear formulas and examples makes geometry practical and easy to apply. Whether you’re designing a box or analyzing materials, knowing how to compute a rectangular prism’s surface area sets a strong foundation.", "---", "Keywords: rectangular prism surface area, formula for rectangular prism surface area, calculate surface area 4x5x6 cm, geometry problem solution, 3D shape surface area, cm² calculator, 4 cm × 5 cm × 6 cm prism, total surface area of rectangular prism", "---", "Ready to master more geometry? Explore related concepts like volume calculations or diagonal measurements next!"]
