Calculation of the volume of an ellipse cone

[Byron Rogers]

I am trying to work out a formula for the approximate calculation of the lung capacity of a racehorse.

http://performancegenetics.com/wp-content/uploads/2015/05/Horse.jpg

I take three physical dimensions on the horse.

1) The measurement of the girth (which is the perimeter of an ellipse) around the horse.
2) the measurement from the top of the withers which coincides with the top of the girth to the point of the hip (B to C)
3) the measurement from the bottom of the girth to the point of the hip (A to C)

Some real data

Girth Circumference (perimeter of Ellipse) - 172 cm
Distance B to C - 65 cm
Distance A to C - 85 cm

The minor axis and major axis ratio’s are approximations that I have made after measuring a few 100 horses.

Major Axis is B to C divided by 2.25 (in this case 76.44)
Minor Axis is A to C divided by 5.75 (in this case 29.91)

I tried using the Heron formula to calculate the height of the triangle and then calculated the Area of the Ellipse ((Pi*Major*Minor)/4) and then calculated the Volume ((base*height)/3)

The volume figures that I am coming up with don't correspond to physical calculations of lung volume in the veterinary world. I am not expecting it to be perfectly accurate, just an approximation, but the figures I am getting are too far away from reality.

 

[Byron Rogers]

I should have added that the cone is obviously not an equilateral cone/triangle.

Any help most appreciated.

 

[Ray Vickson]

I should have added that the cone is obviously not an equilateral cone/triangle.

Any help most appreciated.

Standard formula: $V = \frac{1}{3} A h$, $A=$ area of base, $h =$ height. This is the volume formula for a pyramid, but the bottom need not be a rectangle and the thing can be tilted, as is yours. The reason is simple: the area $a(x)$ at a height $x$ from the apex is $a(x) = A (x/h)^2$ (because the lateral dimensions are proportional to $x$, so the area is proportional to $x^2$) Now integrate: $V = \int_0^h a(x) \, dx.$

 

[Byron Rogers]

Standard formula: $V = \frac{1}{3} A h$, $A=$ area of base, $h =$ height. This is the volume formula for a pyramid, but the bottom need not be a rectangle and the thing can be tilted, as is yours. The reason is simple: the area $a(x)$ at a height $x$ from the apex is $a(x) = A (x/h)^2$ (because the lateral dimensions are proportional to $x$, so the area is proportional to $x^2$) Now integrate: $V = \int_0^h a(x) \, dx.$

Ray. Appreciate the feedback. Is there an easy way to write this in an Excel cell?

 

[Ray Vickson]

Ray. Appreciate the feedback. Is there an easy way to write this in an Excel cell?

The "height" $h$ is the perpendicular distance from your apex C to the plane containing the ellipse AB. The area of an ellipse is $\cal{A} = \pi a b$, where $a =$ major semiaxis and $b =$ minor semiaxis (that is, the length of the major axis is $AB = 2a$ and the length of the minor axis is $2b$). [Note: in my previous post I should really have used $\cal{A}$ instead of $A$ to denote area, because you are already using the letter A as a point-label.) If you know the distances AB, AC and BC you can use standard geometry to find angles, and from that you can find the height $h$. Google 'triangle geometry' or 'triangle formulas', or something similar, to find the appropriate formulas.