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
θ = half-angle or apex angle of the full cone between the center axis and any side, in degrees
φ = 2θ = aperture or opening angle or vertex angle of the full cone between opposite sides, in degrees
β = base angle of the cone, in degrees
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 Angles of the Full Cone

The conical frustum is the bottom section of a cone. If you look straight on to a full cone in two dimensions you will see an isosceles triangle. If you draw the height line from the center of the apex down to bisect the base you will get two right triangles formed by the height h, the radius r on the bottom, and the side s. It can be shown that the angle at the top of the right triangle, called the half-angle, is given by θ = arctan(r/h). If you know θ, and you know you have one right angle at 90°, then you can calculate the third, base angle β = 180 - 90 - θ.

Similarly, look at the frustum straight on in two dimensions. Slide the height line h to the right until it touchs the top of the side s. You form a small triangle that is congruent to the right triangle formed for the full cone. The angles will also be congruent. The new location of where h touches the base is given by r2 - r1. Therefore, for a frustum, you can calculate θ = arctan((r2 - r1)/h). And it is still true that base angle β = 180 - 90 - θ.

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.