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.
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.
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.
Use it to validate homework steps, check CAD or fabrication dimensions, estimate material requirements, and sanity-check hand calculations before submitting work.
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)R²H
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.
This calculator combines the core geometry formula with the input mode selected in the interface, then derives companion values shown in the output cards, comparison bars, and reference tables. Use it to cross-check both direct calculations and reverse-solving scenarios where one measurement is unknown.
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.
Keep units consistent across every input before interpreting area, perimeter, angle, or volume outputs. For best results, measure carefully, round only at the final step, and compare at least one manual calculation with the calculator output when building confidence.
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. Use this as a practical reminder before finalizing the result.
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. Keep this note short and outcome-focused for reuse.