A pyramid with a square base of side 10 cm and height 15 cm is cut by a plane parallel to the base at half its height. What is the volume of the smaller, lower pyramid formed?

["Title: Volume of the Smaller Pyramid Formed by a Parallel Cut at Half the Height", "Meta Description:\nDiscover how to calculate the volume of a smaller, lower pyramid formed when a square-based pyramid is cut parallel to its base at half its height. Learn step-by-step with geometric formulas and real-world applications.", "---", "### Understanding the Original Pyramid", "Imagine a pyramid with a square base where each side measures 10 cm and the total height is 15 cm. This classic geometric figure is not only visually striking but also mathematically rich. To understand the volume of the smaller pyramid formed below, we first calculate the volume of the original pyramid, then analyze how scaling affects its dimensions and volume when sliced at half the height.", "---", "### Step 1: Volume of the Original Pyramid", "The volume ( V ) of a pyramid is given by:\n[\nV = \frac{1}{3} \ imes \ ext{Base Area} \ imes \ ext{Height}\n]", "For our pyramid:\n- Base side length = 10 cm → base area = ( 10 \ imes 10 = 100 , \ ext{cm}^2 )\n- Height = 15 cm", "So,\n[\nV_{\ ext{original}} = \frac{1}{3} \ imes 100 \ imes 15 = 500 , \ ext{cm}^3\n]", "---", "### Step 2: Geometry of the Smaller, Lower Pyramid", "The pyramid is cut by a plane parallel to the base at half its height (7.5 cm). Because the cut is parallel, the smaller pyramid formed at the base is similar to the original pyramid.", "Similarity Principle:\nWhen two pyramids are similar (same shape, different size), the ratio of their volumes equals the cube of the ratio of their corresponding linear dimensions (heights, base edges, etc.).", "Since the cut occurs at half the original height, the ratio of heights is:\n[\n\frac{h_{\ ext{small}}}{h_{\ ext{original}}} = \frac{7.5}{15} = \frac{1}{2}\n]", "Therefore, all linear dimensions of the smaller pyramid are scaled by ( \frac{1}{2} ):\n- Side of base = ( 10 \ imes \frac{1}{2} = 5 , \ ext{cm} )\n- Height = ( 15 \ imes \frac{1}{2} = 7.5 , \ ext{cm} )", "Base area of smaller pyramid:\n[\n5 \ imes 5 = 25 , \ ext{cm}^2\n]", "---", "### Step 3: Volume of the Smaller Pyramid", "Using the pyramid volume formula:\n[\nV_{\ ext{smaller}} = \frac{1}{3} \ imes 25 \ imes 7.5 = \frac{1}{3} \ imes 187.5 = 62.5 , \ ext{cm}^3\n]", "Alternatively, since volume scales with the cube of the height ratio:\n[\nV_{\ ext{smaller}} = V_{\ ext{original}} \ imes \left( \frac{1}{2} \right)^3 = 500 \ imes \frac{1}{8} = 62.5 , \ ext{cm}^3\n]", "---", "### Practical Implications and Applications", "This geometric principle is not just theoretical. In architecture, sculpture, and packaging design, understanding how cutting dimensions scale volumes helps predict material requirements and structural stability. For example, reducing a pyramid’s height to half reduces its volume to one-eighth—critical when replicating proportional models.", "---", "### Summary", "- Original pyramid: base = 10 cm, height = 15 cm → volume = 500 cm³\n- Cut at half-height → smaller pyramid has linear scale factor of 1/2\n- Smaller pyramid volume: ( 500 \ imes \frac{1}{8} = 62.5 , \ ext{cm}^3 )", "This elegant demonstration of geometric similarity reinforces how proportional cuts shape volume, offering valuable insights for math education, engineering, and design.", "---", "Keywords: pyramid volume, square base pyramid, geometry, similar pyramids, volume of smaller pyramid, scaling factor, cube of ratio, half height cut, 10 cm base, 15 cm height, pyramid cut parallel to base", "For more insights into geometric volumes and practical measurements, explore related content on shaping principles in architecture and design."]