Cone Frustum Height from Volume and Outer Angle

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Dec 26, 2024, 1:55:59 PM

Created by

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Dec 23, 2024, 1:49:02 PM

= Cone Frustum Height from Volume and Outer Angle

$= "Cone Frustum Height from Volume and Outer Angle"$

(α) Side Angle

$(alpha)"Side Angle"$

(r) Radius of Bottom

$(r)"Radius of Bottom"$

(V) Volume of Frustum

$(V) "Volume of Frustum"$

The Math

The frustum is the difference of two cones where:

the smaller cone has radius 'r' and height 'a'
the larger cone has radius 'R' and height 'A'

and we are seeking h=A-a.

The larger cone has volume (PI/3)*A*R*R; the smaller was volume (PI/3)*a*r*r. We are given the frustum volume (V) which is the difference of the two. We are also given the angle alpha, thus we know:

a = r * tan(alpha)
A = R * tan(alpha)

Since we know 'r', thus we know 'a'. We can just re-write:

`V = (PI*tan(alpha)/3)*(R^3-r*r*r)

thus we can express 'R' in terms of things we know and we know 'A' is just R * tan(alpha).

The closed form for 'R' is a bit messy:

R = cube root(3*V*cot(alpha)/PI + r*r*r)

Cone Calculators

Cone Surface Area based on cone height and cone base radius
Cone Volume based on cone height and cone base radius
Cone Mass or Weight as a function of the volume and mean density
Cone Frustum Surface Area based on the two heights and two radii
Cone Frustum Volume based on the two heights and two radii
Cone Frustum Weight or Mass based on the cone frustum volume and mean density
Cone Shell Volume a.k.a. hollow cone volume
Cone Shell Mass or Weight a.k.a. hollow cone mass
Moment of inertia of a cone shaped object based on mass and rotation axis
Moment of inertia of a cone shaped object based on mass and distance to apex (rotation around apex)
Compute the total surface are of the frustum of a cone: This includes the top and bottom surfaces.
Compute the lateral surface area (using L) of the frustum of a cone.
Cone Frustum Volume from Angle
Cone Frustum Height from Volume and Outer Side Angle: Computes height based on volume, small radius and outer side angle.
Cone Frustum Height from Volume and Inner Side Angle: Computes height based on volume, small radius and inner side angle.

Volume is a three dimensional measurement of the amount of space taken up by an object. Volume units are cubic measurements for solid objects such as cubic inches and cubic meters. Fluids have separate volume units such as liters, fluid ounces, cups, gallons, and barrel.

The volume of an object can measured by the liquid it displaces or be calculated by measuring its dimensions and applying those dimensions to a formula describing its shape. Many such calculations are available in the following list of calculators.

In many cases, the calculators are for a column with a geometric shaped base and vertical sides. One basic formula for volume is area times a Height when the volume has vertical sides.