Truncated Cone (Frustum) Calculator

Calculate the volume, slant height, lateral surface area, and total surface area of a truncated cone (frustum).

Enter Dimensions

R r h s

Results

Volume
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cubic units

Step-by-Step Solution

V = (pi h / 3)(R^2 + Rr + r^2)

What Is a Truncated Cone (Frustum)?

A truncated cone, also called a frustum, is the shape formed when a cone is cut by a plane parallel to its base. It has two circular faces of different radii (the top and bottom) and a slanted lateral surface connecting them. Frustums appear frequently in engineering, architecture, and everyday objects like buckets, lampshades, and drinking cups.

Frustum Formulas

Volume

The volume of a frustum with bottom radius R, top radius r, and height h.

V = (pi h / 3)(R^2 + Rr + r^2)

Slant Height

The distance along the slanted side from bottom edge to top edge.

s = sqrt(h^2 + (R - r)^2)

Lateral Surface Area

The area of the slanted side surface only.

A_lat = pi (R + r) s

Total Surface Area

Lateral area plus both circular bases.

A_total = A_lat + pi R^2 + pi r^2

Semi-vertical Angle

The angle between the slant and the vertical axis.

alpha = arctan((R - r) / h)

Centroid Height

Height of the center of mass above the base.

y = h(R^2+2Rr+3r^2) / (4(R^2+Rr+r^2))

Practical Applications

Frustum calculations are used in civil engineering (dam walls, foundations, embankments), manufacturing (conical vessels, funnels), and architecture (columns, towers). Understanding frustum geometry helps in calculating material requirements, liquid capacity, and structural properties.

Special Cases

  • When r = 0, the frustum becomes a full cone.
  • When R = r, the frustum becomes a cylinder.
  • The volume formula reduces to V = (pi h / 3) R^2 for a cone (r = 0).
  • The volume formula reduces to V = pi R^2 h for a cylinder (R = r).