Incremental Volume of a Cone Calculator

[civilbill]

Trying to write a formula in excel that will calculate the incremental volume of a cone inch by inch. The diameter of the top of the cone is 34' and the height is 18.75'. Total volume is 5674.50 cuft but I can't figure out how to write the formula so I can generate the volume as I move down the height of the cone. This is generating the volume of solids in the cone. Maybe it is because it is Monday, but I can't figure this out. Thanks.

 

[Ron]

The volume of a cylinder is [[(pi)*(d^2)]/4]*H. A cone is similar to a cylinder but with tapered sides. If you substitute the average diameter from one inch to another [(d1+d2]/2 into the first equation, you will have your answer in increments, using "H" in the increments you choose.

 

[civilbill]

Thanks Ron. Once my coffee kicked in I got it figured out. I wrote a formula that calculated the radius based on the slant height and copied it down the spreadsheet.

 

[Ron]

civilbill..I can relate to that! Every day.

 

[Denial]

Ron,

I don't think you are exactly right. The volume of the cone's frustrum will be slightly larger than your approach predicts, because the volume of a cone is proportional to the cube of the height rather than to the square of the height.

 

[Denial]

Ron again,

A quick bit of algebra suggests that your approach underestimates the frustrum's volume by
pi * (tan(t))^2 * (delta H)^3 / 12
where t is the semi-angle at the cone's apex, and
"delta H" is the thickness of the frustrum.

 

[Ron]

Denial...my approach was a simple approximation of the volume based on the average of the incremental diameter. Considering that the OP wanted an "inch by inch" volume of a cone having a height of 18+ feet, an approximation is appropriate and actually quite accurate...similar to integration. (My volume was 5674.47.....the given volume was 5674.5).

Based on the given volume, base diameter and height, the minor radius is zero, so this is not a frustum, but a true cone.

 

[civilbill]

The method I used was calculating the radius by first calculating the slant height given the slope of the cone and the known height between increments. Once the slant height was known the, the radius was calculated using sqrt(a^2-b^2). Knowing the height and radius, the volume could be calculated and copied down the spreadsheet. Thanks for your help though.