A cone has a height of 9 cm and a base radius of 4 cm. If the cone is sliced parallel to its base at half its height, what is the volume of the smaller cone formed?
["Title: Volume of a Smaller Cone Formed by Cutting a Cone Parallel to Its Base at Half Its Height (9 cm Height, 4 cm Radius)", "When working with cones, understanding how slicing affects shape and volume is essential in geometry, engineering, and design. In this SEO-optimized article, we explore the volume of a smaller cone created when a cone is cut parallel to its base at half its height. Specifically, we analyze a cone with a height of 9 cm and a base radius of 4 cm. After slicing at the midpoint, we calculate the volume of the resulting smaller cone.", "---", "### Understanding a Cone’s Geometry and Volume Formula", "A cone is defined by its height and base radius. The volume ( V ) of a full cone is calculated using the formula:", "[\nV = \frac{1}{3} \pi r^2 h\n]", "Where:\n- ( r ) = radius of the base\n- ( h ) = vertical height from base to apex", "In our example:\n- Height ( h = 9 ) cm\n- Base radius ( r = 4 ) cm", "Substituting into the volume formula:", "[\nV = \frac{1}{3} \pi (4)^2 (9) = \frac{1}{3} \pi \ imes 16 \ imes 9 = \frac{144}{3} \pi = 48\pi \ ext{ cm}^3\n]", "So, the original cone has a volume of ( 48\pi ) cm³.", "---", "### Cutting the Cone at Half Its Height", "When the cone is sliced parallel to its base at half its height, the slicing occurs at:", "[\n\frac{9}{2} = 4.5 \ ext{ cm from the base (or 4.5 cm from the base and 4.5 cm from the apex)}\n]", "This slice forms a smaller, similar cone on top — top portion of the original cone — which is geometrically proportional to the full cone.", "Because the cone maintains shape (similar figures), all linear dimensions scale by a constant factor. Since the cut is halfway up the height, the height of the smaller cone is:", "[\nh_{\ ext{small}} = 4.5 \ ext{ cm}\n]", "And due to similarity, the radius scales by the same ratio:", "[\nr_{\ ext{small}} = \frac{4.5}{9} \ imes 4 = 0.5 \ imes 4 = 2 \ ext{ cm}\n]", "---", "### Calculating the Volume of the Smaller Cone", "Now apply the volume formula to the smaller cone with height ( 4.5 ) cm and radius ( 2 ) cm:", "[\nV_{\ ext{small}} = \frac{1}{3} \pi (2)^2 (4.5) = \frac{1}{3} \pi \ imes 4 \ imes 4.5 = \frac{1}{3} \pi \ imes 18 = 6\pi \ ext{ cm}^3\n]", "---", "### Bonus: What About the Remaining Lower Cone?", "While the question focuses on the smaller cone above, it’s worth noting that slicing midway creates:\n- A smaller cone (top) with volume ( 6\pi ) cm³\n- A larger frustum-shaped lower portion (not asked here)", "But mathematically, due to similarity ratios, the volume of the smaller cone formed at half height is always ( \left(\frac{1}{2}\right)^3 = \frac{1}{8} ) of the original cone’s volume.", "[\n\frac{1}{8} \ imes 48\pi = 6\pi \ ext{ cm}^3\n]", "This confirms our result using the scaling law.", "---", "### Practical Applications", "Understanding cone volume partitions is crucial in fields such as:\n- Engineering: Designing conical tanks or funnels\n- Manufacturing: Calculating material volumes in cone-shaped components\n- Architecture: Estimating filled volumes in decorative cones\n- Education: Teaching similarity and proportional reasoning", "---", "### Summary", "Given a cone with:\n- Height = 9 cm\n- Base radius = 4 cm", "When sliced parallel to base at half height (4.5 cm), the resulting smaller cone has:\n- Height = 4.5 cm\n- Radius = 2 cm", "Volume of the smaller cone: 6π cm³ ≈ 18.85 cm³", "This elegant geometric relationship demonstrates how cutting a cone uniformly preserves proportionality and enables precise volume computation.", "---", "Keywords: cone volume, volume of cone, sliced cone volume, cone geometry, radius height relationship, smaller cone volume, geometric similarity, frustum vs cone volume, math education, cone calculations", "Meta Description: Learn how slicing a cone at half its height creates a smaller cone with volume ( 6\pi ) cm³ using proportional geometry and volume formulas. Perfect for math students and educators.", "---", "Always verify dimensions and apply similarity principles when working with scaled geometric shapes to calculate precise volumes—especially in practical applications."]
