Conical Frustum Volume Calculator

A cone is a three-dimensional shape with a circular base that tapers to a single point called the apex or vertex. A conical frustum, on the other hand, is what you get when you cut off the top portion of a cone with a plane parallel to the base, resulting in a shape with two circular ends of different sizes (like a truncated cone). Unlike a full cone, a conical frustum doesn't have a pointed tip. The frustum has two circular faces — a larger bottom base and a smaller top base — connected by a slanted surface. The volume formula for a frustum, V = (1/3)πh(R² + Rr + r²), is derived by subtracting the volume of the removed cone tip from the full cone volume. This shape appears in countless everyday objects.

The volume of a conical frustum (truncated cone) is V = (π × h × (r₁² + r₁ × r₂ + r₂²)) / 3, where h is the vertical height, r₁ is the top radius, and r₂ is the bottom radius. This formula accounts for the varying circular cross-sections. For example, if R = 5 cm, r = 3 cm, and h = 10 cm, then V = (π × 10 × (25 + 15 + 9)) / 3 = (π × 10 × 49) / 3 = 1,540π/3 ≈ 512.7 cm³. The formula works regardless of which radius is larger, as it is symmetric — swapping R and r gives the same result. The term (R² + Rr + r²) represents the weighted average of the top and bottom circular areas.

Yes, the formula V = (1/3) × π × h × (R² + R×r + r²) works regardless of which radius is larger. The calculations are symmetrical with respect to the two radii — swapping the top and bottom radius values produces the same volume result. However, in traditional terminology, the larger circle is usually considered the base (bottom) and the smaller circle is the top. If your conical frustum is positioned with the larger radius at the top (like an inverted bucket), you may want to mentally reorient it for clarity, but the volume calculation remains identical. This symmetry makes the frustum formula very versatile for many applications where the orientation may vary, such as funnels, flower pots, and architectural columns.

If you know the slant height (s) instead of the vertical height (h), you can calculate the vertical height using the Pythagorean theorem applied to the frustum's cross-section: h² = s² - (R - r)², where R is the bottom radius and r is the top radius. Once you've calculated the vertical height using h = √(s² - (R - r)²), you can use it in the standard frustum volume formula. The slant height is the distance along the side of the frustum from the edge of the bottom base to the corresponding edge of the top base. For example, if R = 5 cm, r = 3 cm, and s = 12 cm, then h = √(144 - 4) = √140 ≈ 11.83 cm. This conversion is essential when measuring tapered objects where direct vertical measurement is difficult.

Frustum volumes are used in countless everyday and industrial applications. Common examples include: buckets and pails (a standard 10-liter bucket has a frustum shape for nesting), flower pots (garden pots are conical frustums — a 30cm top diameter, 20cm bottom diameter, 25cm height pot holds ~12.5 liters of soil), lampshades, funnels, concrete hoppers (a hopper with 1m top radius, 0.3m bottom radius, 1.5m height holds ~2.8 m³ of concrete mix), cooling towers at power stations, architectural tapered columns, some water tanks, and tapered pipes or ducts in engineering. Understanding frustum volume is essential for manufacturers, architects, and engineers who work with tapered containers or structural elements.

If the top and bottom radii are equal (r = R), the conical frustum becomes a cylinder, and the formula V = (1/3)πh(R² + Rr + r²) simplifies to V = (1/3)πh(R² + R² + R²) = (1/3)πh(3R²) = πR²h, which is the standard cylinder volume formula. Conversely, if the top radius approaches zero (r → 0), the frustum approaches a complete cone, and the formula simplifies to V = (1/3)πR²h, the standard cone volume formula. This mathematical elegance shows how the frustum formula naturally bridges the cylinder and cone formulas — it's the general case that encompasses both special cases. Our calculator handles all these scenarios automatically.