Question: A right circular cone has a height of $12$ units and a base radius of $5$ units. A smaller cone is cut from the top, leaving a frustum with a height of $9$ units. What is the volume of the frustum?
["What’s the Volume of a Frustum When a Cone Is Cut from the Top? Insights for Clear Understanding", "How many times have you wondered about shaping space in math class, only to pause when exploring real-world geometry? A question frequently surface in US classrooms and on digital learning platforms: A right circular cone has a height of 12 units and a base radius of 5 units. A smaller cone is cut from the top, leaving a frustum with a height of 9 units. What is the volume of the frustum? This isn’t just a textbook problem—understanding frustum volume reveals how shape, space, and design intersect in engineering, architecture, and everyday innovation.", "This query aligns with growing interest in applied geometry, especially in educational tech environments where visual learning and spatial reasoning are emphasized. Users searching for clear, practical explanations—whether students, educators, or design enthusiasts—seek reliable methods to calculate volume without confusion.", "### The Science Behind the Frustum Volume Formula", "A frustum is the tapered portion remaining when a cone loses its tip. Its volume can be derived cleanly from basic cone formulas. Recall that the volume \( V \) of a full cone is \( \frac{1}{3} \pi r^2 h \). Here, the original cone stands 12 units tall with base radius 5. The smaller cone removed shares the same proportions—its height and radius shrink proportionally from the top.", "Since the remaining frustum is 9 units tall, the smaller cone’s height must be \( 12 - 9 = 3 \) units. Because the cones are similar, the radius of the smaller cone scales with height: radius scales 3/12 = 1/4, so the smaller cone’s radius is \( 5 \ imes \frac{1}{4} = 1.25 \) units.", "Calculating volumes: \n- Original cone: \( \frac{1}{3} \pi (5)^2 (12) = \frac{1}{3} \pi (25)(12) = 100\pi \) \n- Smaller cone: \( \frac{1}{3} \pi (1.25)^2 (3) = \frac{1}{3} "]
