Skip to main content
K
KenFarley21-Topaz II
21-Topaz II
March 28, 2014
Solved

Model with simple frustrum - volume calculation wrong

  • March 28, 2014
  • 1 reply
  • 7746 views

I make a lot of models that involve pyramidal frustrums (transition of a rectangular section to another rectangular section, linearly). Rather than use the measured value calculated by Creo, I thought I'd have a go at calculating the volume directly from the dimensions.

When I check my calculations, the volume calculated directly from the dimensions differs from the Creo analysis result.

I minimized the accuracy value, but still get the exact same result.

Here are my simple values

Base rectangle: 5 X 3 -> Base area: 15

Top rectangle: 4 X 2 -> Top area: 8

Height: 1

Frustrum of a pyramid volume is

Volume = ( Area Base + Area Top + SQRT ( Area Base * Area Top ) ) * Height / 3 = 11.3182

Creo, on the other hand, gives a volume of 11.3333.

Anyone have any insights into this?

This is simple shape, and kind of scares me about what might be happening with more complicated geometry.

    Best answer by dschenken

    I think Creo is right.

    I'm not sure where the formula came from but ...

    L = l_base - ((l_base-l_top)/delta_z) * Z

    W = w_base - ((w_base-w_top)/delta_z) * Z

    A = L * W =

    l_base*w_base

    - (l_base*((w_base-w_top)/delta_z) + w_base*((l_base-l_top)/delta_z)) * Z

    + ((l_base-l_top)/delta_z)*((w_base-w_top)/delta_z)*Z*Z

    Integrating:

    V =

    l_base*w_base*Z

    - (l_base*((w_base-w_top)/delta_z) + w_base*((l_base-l_top)/delta_z)) * Z*Z/2

    + ((l_base-l_top)/delta_z)*((w_base-w_top)/delta_z)*Z*Z*Z/3

    Plugging it all in I get 11.33...

    1 reply

    D
    dschenken1-VisitorAnswer
    1-Visitor
    March 28, 2014

    I think Creo is right.

    I'm not sure where the formula came from but ...

    L = l_base - ((l_base-l_top)/delta_z) * Z

    W = w_base - ((w_base-w_top)/delta_z) * Z

    A = L * W =

    l_base*w_base

    - (l_base*((w_base-w_top)/delta_z) + w_base*((l_base-l_top)/delta_z)) * Z

    + ((l_base-l_top)/delta_z)*((w_base-w_top)/delta_z)*Z*Z

    Integrating:

    V =

    l_base*w_base*Z

    - (l_base*((w_base-w_top)/delta_z) + w_base*((l_base-l_top)/delta_z)) * Z*Z/2

    + ((l_base-l_top)/delta_z)*((w_base-w_top)/delta_z)*Z*Z*Z/3

    Plugging it all in I get 11.33...

    D
    dschenken1-Visitor
    1-Visitor
    March 29, 2014

    I should be able to simplify the integral to be the same as the original formula, but so far no luck. I looked at the Wikipedia page that showed the derivation of the original formula, but it doesn't yield terms I can yet convert.

    I get 11.3182 from the original formula, but don't see the reason for the discrepancy.

    D
    dschenken1-Visitor
    1-Visitor
    March 29, 2014

    I put the integration solution and original into Excel and tried some experiments.

    Case 1: the top and bottom the same; the original formula and mine agree.

    Case 2: the top dimensions both zero; the original formula and mine agree.

    Case 3: the top length equal to the bottom and the top width to zero the formulas disagree. Since it's simulating a triangle section, the volume should be 1/2 that of Case1. Mine is, but the original formula gives 1/3.

    What is interesting is that the original formula doesn't distinguish between the case where the top surface area is zero because it is a point and the case where it is a line.

    It may be that the original formula is built on the assumption that the sides all could extend to a single point and is shape independent while the integration handles sides of differing slopes, but only rectangles.

    Terms & ConditionsCookie settingsAccessibility statement