A frustum of a cone has a top radius of 3 cm, a bottom radius of 5 cm, and a height of 7 cm. What is its volume?
["Title: How to Calculate the Volume of a Frustum of a Cone – A Clear Guide", "Meta Description:\nLearn how to calculate the volume of a frustum of a cone using real-world dimensions. This guide explains the formula and computes the volume for a cone with top radius 3 cm, bottom radius 5 cm, and height 7 cm.", "---", "### Introduction", "Geometry plays a crucial role in architecture, engineering, and design — and one fascinating shape is the frustum of a cone. Whether modeling a decorative vase, calculating storage volumes, or planning industrial structures, understanding how to compute the volume of a frustum is essential.", "In this article, we’ll explore the formula for the volume of a frustum of a cone and apply it to a specific example: a frustum with a top radius of 3 cm, bottom radius of 5 cm, and a height of 7 cm.", "---", "### What Is a Frustum of a Cone?", "A frustum is a portion of a cone formed by slicing off the top with a plane parallel to the base. Its geometric properties differ from a full cone, but its volume can still be calculated using a well-established mathematical formula.", "---", "### The Volume Formula", "The volume ( V ) of a frustum of a cone is given by:", "[\nV = \frac{1}{3} \pi h (r_{\ ext{top}}^2 + r_{\ ext{bottom}}^2 + r_{\ ext{top}} \cdot r_{\ ext{bottom}})\n]", "where:\n- ( h ) = vertical height of the frustum\n- ( r_{\ ext{top}} ) = radius of the top circular face\n- ( r_{\ ext{bottom}} ) = radius of the bottom circular face\n- ( \pi \approx 3.1416 )", "---", "### Apply the Formula to the Given Dimensions", "From the problem:", "- Top radius (( r_{\ ext{top}} )) = 3 cm\n- Bottom radius (( r_{\ ext{bottom}} )) = 5 cm\n- Height (( h )) = 7 cm", "Substitute these values into the formula:", "[\nV = \frac{1}{3} \pi \ imes 7 \ imes \left( 3^2 + 5^2 + 3 \ imes 5 \right)\n]", "Calculate step-by-step:", "1. ( 3^2 = 9 )\n2. ( 5^2 = 25 )\n3. ( 3 \ imes 5 = 15 )\n4. Sum: ( 9 + 25 + 15 = 49 )\n5. Multiply by height: ( 49 \ imes 7 = 343 )\n6. Volume: ( V = \frac{1}{3} \pi \ imes 343 = \frac{343}{3} \pi )", "Now compute numerically:", "[\nV \approx \frac{343}{3} \ imes 3.1416 \approx 114.333 \ imes 3.1416 \approx 359.19 \ ext{ cm}^3\n]", "---", "### Result", "The volume of the frustum of a cone with top radius 3 cm, bottom radius 5 cm, and height 7 cm is approximately:", "[\n\ ext{Volume} \approx 359.2 \ ext{ cm}^3\n]", "---", "### Why This Matters", "Understanding frustum volume helps professionals estimate material needs, optimize space, and design conical structures safely and efficiently. Whether you're building a cone-shaped reservoir or crafting a decorative piece, knowing how to calculate volume ensures accuracy and precision.", "---", "### Final Thoughts", "The frustum of a cone may seem mathematically complex at first, but with the right formula and step-by-step substitution, calculating its volume becomes straightforward. The formula combines key geometric elements—heights, radii, and a volume constant—to deliver a powerful tool for real-world applications.", "Ready to calculate? Use this formula confidently with your own measurements — the frustum volume is just a step away!", "---", "Keywords: frustum of a cone, volume of frustum, formula frustum volume, conical geometry, frustum calculation, geometry tutorial, cone volume, mathematical formula, practical geometry, real-world volume calculation"]
