Cone Volume Calculator
Calculate the volume, surface area, lateral area, and slant height of a cone. Supports multiple solve modes, unit conversion, and real-world presets.
Frustum of a Cone Calculator (Volume & Surface Area)
Calculate the volume, lateral surface area, total surface area, and slant height of a truncated cone (frustum). Two radii + height or slant height, with real-world presets.
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Planning notes, formulas, and examples
About the Frustum of a Cone Calculator (Volume & Surface Area)
A frustum of a cone (conical frustum) is the solid shape created when a cone is sliced by a plane parallel to its base, removing the smaller "cap" at the top. The result is a truncated cone with two circular bases of different radii — like a bucket, lamp shade, paper cup, or traffic cone.
The frustum is defined by three measurements: the bottom (larger) radius R, the top (smaller) radius r, and the perpendicular height h. From these, you can derive the slant height l = √(h² + (R−r)²), which is the distance along the sloping side from one base rim to the other.
The volume of a frustum is V = (πh/3)(R² + Rr + r²). This elegant formula interpolates between a cylinder (when R = r) and a full cone (when r = 0). The lateral (side) surface area is π(R + r)l, and the total surface area adds both circular bases: lateral + πR² + πr².
Frustums are everywhere in real life: buckets, drinking cups, plant pots, lamp shades, cooling towers, and architectural columns all have frustum shapes. In engineering, calculating the volume of a frustum is essential for determining material quantities, fluid capacity, and structural loads. This calculator handles both input modes (height or slant height), provides full cone extension analysis, and includes real-world presets for quick reference.
When This Page Helps
The Frustum of a Cone Calculator (Volume & Surface Area) is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Volume, Lateral Surface Area, Total Surface Area in one pass, with conversions and derived values shown together.
How to Use the Inputs
- Choose input mode: Two Radii + Height, or Two Radii + Slant Height.
- Select a measurement unit.
- Enter the bottom radius (R) and top radius (r), where R > r.
- Enter the height or slant height depending on your chosen mode.
- Or click a preset to load a common frustum shape (bucket, cup, etc.).
- View volume, lateral area, total surface area, and full cone analysis.
- Compare the surface area breakdown visually.
Formula used
Volume: V = (πh/3)(R² + Rr + r²)
Lateral Surface Area: AL = π(R + r)l
Total Surface Area: AT = π(R + r)l + πR² + πr²
Slant Height: l = √(h² + (R − r)²)
Height from slant: h = √(l² − (R − r)²)
Full Cone Height: H = hR / (R − r)
Full Cone Volume: (π/3)R²HExample Calculation
Result: Volume ≈ 17,593 cm³, Lateral Area ≈ 2,553 cm², Total SA ≈ 3,660 cm²
For a bucket with R = 15, r = 12, h = 30: Slant = √(900 + 9) ≈ 30.15. Volume = (π×30/3)(225 + 180 + 144) ≈ 17,593. Lateral = π(15 + 12)×30.15 ≈ 2,553. Total = 2,553 + π×225 + π×144 ≈ 3,660.
Tips & Best Practices
- When r = 0, the frustum becomes a full cone. When R = r, it becomes a cylinder. The frustum formula covers both as special cases.
- The slant height l is always greater than or equal to the height h, with equality only when R = r (cylinder).
- For a bucket or pot, the volume tells you the liquid capacity. Remember: 1,000 cm³ = 1 liter.
- The lateral surface area is what you need for the label or wrapping material — it excludes the top and bottom circles.
- A frustum can also be formed from a pyramid — the formulas are analogous but use polygonal bases.
How This Frustum of a Cone Calculator (Volume & Surface Area) Works
Where It Helps In Practice
Frustum of a Cone Calculator (Volume & Surface Area) calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.
Accuracy And Setup Tips
Sources & Methodology
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Frequently Asked Questions
A frustum is the portion of a cone between two parallel planes cutting it. It has a larger circular base (radius R), a smaller top (radius r), and a height h.
V = (πh/3)(R² + Rr + r²). This is a weighted average of the top area, bottom area, and a geometric mean term.
Slant height l = √(h² + (R−r)²), where h is the perpendicular height and R−r is the difference in radii.
Lateral area = π(R + r)l, where l is the slant height. This is the area of the sloping side only, excluding the two bases.
A frustum is a cone with the top cap removed. The full cone height H = hR/(R−r). The frustum volume equals the full cone volume minus the removed cap volume.
Buckets, drinking cups, flower pots, lamp shades, traffic cones, cooling towers, and architectural columns are all frustum shapes.
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