Frustum (Truncated Cone) Surface Area Calculator

Calculate the lateral and total surface area of a frustum (truncated cone) with step-by-step solutions.

Select Input Method & Enter Dimensions

Result

Total Surface Area
--
square units
Lateral Surface Area --
Top Base Area --
Bottom Base Area --
Slant Height --

Step-by-Step Solution

Lateral = pi(R + r)l | Total = Lateral + piR^2 + pir^2

Understanding Frustum Surface Area

A frustum (also called a truncated cone) is the portion of a cone that lies between two parallel planes cutting through it. When a right circular cone is cut by a plane parallel to its base, the shape between the base and the cutting plane is called a frustum. Frustums appear frequently in engineering, architecture, and manufacturing.

The surface area of a frustum consists of three parts: the lateral (slant) surface, the top circular base, and the bottom circular base. Depending on the application, you may need only the lateral area (e.g., for wrapping material) or the total area including both bases.

Frustum Surface Area Formulas

Lateral Surface Area

The curved surface between the two circular ends of the frustum.

A_lateral = pi(R + r) x l

Total Surface Area

The lateral area plus both circular base areas combined.

A_total = pi(R + r)l + piR^2 + pir^2

Slant Height from Vertical Height

If only the vertical height is known, compute slant height first.

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

How to Calculate Frustum Surface Area

Method 1: Using Slant Height Directly

  1. Measure the large radius (R) of the bottom base.
  2. Measure the small radius (r) of the top base.
  3. Measure the slant height (l) along the lateral surface.
  4. Compute lateral area: A_lateral = pi(R + r) x l.
  5. Add base areas for total: A_total = pi(R + r)l + piR^2 + pir^2.

Method 2: Using Vertical Height

  1. Measure R, r, and the vertical height h between the bases.
  2. Compute the slant height: l = sqrt(h^2 + (R - r)^2).
  3. Then use the lateral and total area formulas as above.

Practical Applications

Frustum calculations are essential in many real-world situations:

  • Manufacturing: Designing tapered containers, buckets, lampshades, and funnels.
  • Civil Engineering: Designing bridge piers, dam spillways, and tapered columns.
  • Architecture: Calculating material for conical roofs, cooling towers, and decorative elements.
  • Packaging: Determining the material needed for tapered cups and boxes.

Tips for Accurate Calculations

  • Make sure R is always the larger radius and r the smaller one.
  • Slant height is measured along the surface, not straight up through the center.
  • If you know the vertical height, use the Pythagorean theorem to find slant height.
  • Keep all measurements in the same unit before calculating.