A rectangular prism has dimensions 8 cm by 6 cm by 4 cm. If a cylindrical hole with a radius of 1 cm is drilled through the center along the longest dimension, what is the remaining volume?
["### Remaining Volume After Drilling a Cylindrical Hole Through a Rectangular Prism", "When working with geometric shapes in engineering, architecture, or education, calculating the remaining volume after material removal is essential. This SEO-optimized article explains how to determine the remaining volume when a cylindrical hole is drilled through a rectangular prism.", "---", "#### Given Dimensions\nWe start with a rectangular prism (also known as a rectangular parallelepiped) with dimensions:", "- Length: 8 cm (the longest dimension)\n- Width: 6 cm\n- Height: 4 cm", "This prism has a total volume calculated by multiplying its length, width, and height:", "[\n\ ext{Volume}{\ ext{prism}} = 8 , \ ext{cm} \ imes 6 , \ ext{cm} \ imes 4 , \ ext{cm} = 192 , \ ext{cm}^3\n]", "---", "#### Drilling a Cylindrical Hole\nA cylindrical hole with radius ( r = 1 , \ ext{cm} ) is drilled straight through the center along the 8 cm length. This means:", "- The height of the cylinder equals the prism’s longest dimension: ( h = 8 , \ ext{cm} )\n- The cylinder’s radius is ( 1 , \ ext{cm} ), giving a diameter of ( 2 , \ ext{cm} ), which fits perfectly across the 6 cm width (allowing space on both sides).", "The volume of the removed cylindrical material is:", "[\n\ ext{Volume}^3}} = \pi r^2 h = \pi (1, \ ext{cm})^2 (8, \ ext{cm}) = 8\pi , \ ext{cm\n]", "---", "#### Remaining Volume Calculation", "Subtract the cylinder’s volume from the original prism volume:", "[\n\ ext{Remaining Volume} = 192 , \ ext{cm}^3 - 8\pi , \ ext{cm}^3\n]", "Using ( \pi \approx 3.1416 ), calculate ( 8\pi \approx 25.133 ), so:", "[\n\ ext{Remaining Volume} \approx 192 - 25.133 = 166.867 , \ ext{cm}^3\n]", "For exact implementation, leave the volume in symbolic form:", "[\n\boxed{192 - 8\pi , \ ext{cm}^3}\n]", "---", "#### Practical Significance", "Understanding the remaining volume after hole drilling helps in material estimation, weight calculations, structural integrity analysis, and design optimization. This principle applies in manufacturing, 3D printing, and construction—making precise volume calculations crucial for efficient resource use and accuracy.", "---", "Summary:\n- Original prism volume: ( 192 , \ ext{cm}^3 )\n- Cylindrical hole volume: ( 8\pi , \ ext{cm}^3 )\n- Remaining volume: ( 192 - 8\pi , \ ext{cm}^3 ) (≈ 166.87 cm³)", "This approach ensures accurate volume assessment after structural modifications.", "---", "Keywords: rectangular prism volume, cylindrical hole volume, remaining volume calculation, geometry, 3D shape computation, engineering volume, industrials design, volume after drilling, prism with cylindrical cavity"]
