Enter the bottom radius (R), top radius (r), and height (h) of your truncated cone to calculate its volume, lateral surface area, total surface area, and slant height. The Truncated Cone (Frustum) Calculator applies standard frustum geometry formulas, giving you all key measurements in one step.
Volume (V)
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Slant Height (s)
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Lateral Surface Area (L)
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Top Surface Area (T)
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Base Surface Area (B)
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Total Surface Area (A)
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A frustum is what you get when you cut the top off a right circular cone with a plane parallel to its base. The result is a shape with two circular faces of different radii connected by a slanted lateral surface. Common real-world examples include buckets, lampshades, and cups.
Use the formula V = (1/3) × π × h × (R² + R×r + r²), where R is the bottom radius, r is the top radius, and h is the height. This formula derives from subtracting the volume of the small removed cone from the original full cone's volume.
Plugging into V = (1/3) × π × 5 × (4 + 2 + 1) = (1/3) × π × 5 × 7 ≈ 36.65 cm³. You can verify this instantly using the calculator above by entering R = 2, r = 1, and h = 5.
The slant height (s) is the distance along the lateral surface from the edge of the bottom circle to the edge of the top circle. It is calculated as s = √(h² + (R − r)²), using the Pythagorean theorem on the right triangle formed by the height and the difference in radii.
The lateral surface area covers only the slanted outer wall, not the two circular ends. The formula is L = π × (R + r) × s, where s is the slant height. It represents the area you would need to cover if wrapping the side of the frustum.
Lateral surface area (L) covers only the slanted side wall. Total surface area (A) adds the areas of both circular bases: A = L + π×R² + π×r². If you need to paint or cover an entire frustum including top and bottom, use the total surface area.
Frustum calculations are used in engineering, construction, and manufacturing whenever tapered shapes appear — such as designing grain silos, water tanks, traffic cones, architectural columns, sheet-metal funnels, or even cosplay props where you need to cut and roll flat material into a cone shape.
The formulas treat R as the bottom (larger) radius and r as the top (smaller) radius, but mathematically the results for volume and surface area are symmetric — swapping R and r gives the same values. The slant height and all areas depend on the difference and sum of the radii, not their order.