Calculators

Geometry

Solid

Conical Frustum Shape
(of right circular cone)

r₁ = radius1
r₂ = radius2
h = height
s = slant height
V = volume
L = lateral surface area
T = top surface area
B = base surface area
A = total surface area
π = pi = 3.1415926535898
√ = square root

Calculator Use

This online calculator will calculate the various properties of a conical frustum given the 2 radii and any 1 other known variable. This geometric solid conical frustum is a type of right circular cone, where a right cone is a cone with its vertex point above the center of its base. The frustum is a cone with the top cut off by making a slice parallel to the base. Answers will include a link to the calculation of the full cone.

Note that frustum is often misspelled "frustrum."

Units: Note that units are shown for convenience but do not affect the calculations. The units are in place to give an indication of the order of the results such as ft, ft² or ft³. For example, if you are starting with mm and you know r and h in mm, your calculations will result with s in mm, V in mm³, L in mm², T in mm², B in mm² and A in mm².

Below are the standard formulas for a conical frustum. Calculations are based on algebraic manipulation of these standard formulas.

Conical Frustum Formulas in terms of r and h:

• Slant height of a conical frustum:
  • s = √((r₁ - r₂)² + h²)
• Volume of a conical frustum:
  • V = (1/3) * π * h * (r₁² + r₂² + (r₁ * r₂))
• Lateral surface area of a conical frustum:
  • S = π * (r₁ + r₂) * s = π * (r₁ + r₂) * √((r₁ - r₂)² + h²)
• Top surface area of a conical frustum (a circle):
  • T = πr₁²
• Base surface area of a conical frustum (a circle):
  • B = πr₂²
• Total surface area of a conical frustum:
  • A = π * (r₁² + r₂² + (r₁ + r₂) * s) = π * [ r₁² + r₂² + (r₁ + r₂) * √((r₁ - r₂)² + h²) ]

Conical Frustum Calculations:

Use the following additional formulas along with the formulas above.

• Given radius1, radius2 and height calculate the slant height, volume, lateral surface area and total surface area.
  Given r₁, r₂, h find s, V, S, A
  • use the formulas above
• Given radius1, radius2 and slant height calculate the height, volume, lateral surface area and total surface area.
  Given r₁, r₂, s find h, V, S, A
  • h = √(s² - (r₁ - r₂)²)
• Given radius1, radius2 and volume calculate the height, slant height, lateral surface area and total surface area.
  Given r₁, r₂, V find h, s, S, A
  • h = (3 * V) / (π * (r₁² + r₂² + (r₁ * r₂)))
• Given radius1, radius2 and lateral surface area calculate the height, slant height, volume and total surface area.
  Given r₁, r₂, S find h, s, V, A
  • s = S / (π * (r₁ + r₂))
  • h = √(s² - (r₁ - r₂)²)
• Given radius1, radius2 and total surface area calculate the height, slant height, volume and lateral surface area.
  Given r₁, r₂, A find h, s, V, S
  • s = [A/π - r₁² - r₂²] / (r₁ + r₂)
  • h = √(s² - (r₁ - r₂)²)

Calculate Cone Shape from the Conical Frustum

If the calculation can be done the answer will include a link to the Cone Calculator presenting the full cone of the expanded frustum.

You can calculate the full volume of the cone, and other properties, using the ASS and ASA triangle theorems.

References

Weisstein, Eric W. "Conical Frustum." From MathWorld--A Wolfram Web Resource. Conical Frustum.