Frustum Calculator
Volume and surface area of a conical frustum.
Volume
527.7876
conical frustum
Lateral surface
241.576
Slant height
8.544
Total surface
382.9477
AI Breakdown & Smart Takeaway
Plain-English insight on your numbers
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How the Frustum Calculator works
The Frustum Calculator computes the volume and lateral and total surface area of a conical frustum (truncated cone) from its top radius, bottom radius, and height — making it indispensable for engineers, architects, students, and makers working with tapered 3D shapes.
A frustum is what you get when you slice a cone with a plane parallel to its base, removing the pointed top and keeping the trapezoidal solid between the two circular faces. Unlike a full cone, a frustum has two circular ends of different radii, which is exactly the geometry found in everyday objects like buckets, grain silos, paper cups, lamp shades, and concrete piers. This calculator asks you for three values — the bottom radius (R), the top radius (r), and the perpendicular height (h) — and instantly returns the volume, lateral surface area, and total surface area.
The volume of a frustum is derived by subtracting the volume of the small removed cone from the volume of the full original cone. This algebraic simplification yields the elegant formula V = (π·h/3)·(R² + R·r + r²), which handles all cases: when r approaches 0 it reduces to the standard cone formula, and when r equals R it reduces to the formula for a cylinder. Understanding this derivation helps you catch input errors — for instance, if your computed volume is unreasonably small, you may have swapped the height for a slant height.
Surface area comes in two flavors. The lateral surface area covers only the sloping side and equals π·(R + r)·l, where l is the slant height computed as √(h² + (R − r)²). The total surface area adds both circular end caps: A_total = π·(R + r)·l + π·R² + π·r². A common mistake is confusing the perpendicular height with the slant height — the calculator uses perpendicular height internally and derives the slant height automatically, so always enter the straight vertical distance between the two bases, not the length along the sloping surface.
Practical applications often involve unit consistency. If you measure radii in centimeters and height in meters, your output will be meaningless; make sure all three inputs share the same unit before calculating. For manufacturing or construction contexts, the lateral surface area directly tells you how much sheet material (metal, fabric, or concrete formwork) you need to wrap the sloping side, while the volume drives material-fill calculations such as how many liters of liquid a tapered tank holds. Always add a small waste factor (typically 5–10 %) to material estimates derived from surface area.
Formula
V = ⅓·π·h·(R² + Rr + r²)
Pro tips
- Always enter the perpendicular (vertical) height, not the slant height — mixing these up is the single most common error and will give incorrect volume and surface area results.
- If you only know the slant height from a physical measurement, first recover the perpendicular height with h = √(l² − (R − r)²) before entering values into the calculator.
- For material estimation, use the lateral surface area output and multiply by your sheet material's thickness to get volume of material needed for the shell — then convert to weight using the material's density.
- When R and r are very close in value (a nearly cylindrical frustum), double-check your measurements carefully because small errors in radius have an outsized percentage impact on the difference (R − r) used in the slant height calculation.
- To find the height of the full original cone before it was truncated, use the relationship H = h · R / (R − r) — useful when designing tapered molds or checking if a fabricated frustum matches its design intent.
Key terms
- Frustum
- — The portion of a solid (typically a cone) that lies between two parallel cutting planes, resulting in a shape with two flat circular faces of different sizes.
- Slant Height (l)
- — The straight-line distance along the sloping surface of the frustum from the edge of the top circle to the edge of the bottom circle, calculated as √(h² + (R − r)²).
- Perpendicular Height (h)
- — The vertical distance measured at a right angle between the two parallel circular bases of the frustum, distinct from the slant height.
- Lateral Surface Area
- — The area of only the curved, sloping outer surface of the frustum, excluding both circular end caps.
- Total Surface Area
- — The complete surface area of the frustum, including the lateral surface plus both circular top and bottom faces.
- Truncated Cone
- — Another name for a conical frustum, referring to a cone whose apex has been cut off by a plane parallel to the base.
