# Calculation of the volume of an ellipse cone

• Byron Rogers
In summary, the conversation discusses the process of trying to work out a formula for calculating the lung capacity of a racehorse based on its physical dimensions. The formula involves measuring the girth, distance from the top of the withers to the point of the hip, and distance from the bottom of the girth to the point of the hip. The conversation also explores using the Heron formula and the volume formula for a pyramid to find the lung capacity, but the results are not accurate. The conversation concludes with the suggestion to use triangle geometry to find the height and ultimately calculate the lung capacity.
Byron Rogers
I am trying to work out a formula for the approximate calculation of the lung capacity of a racehorse.

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.

Last edited by a moderator:
I should have added that the cone is obviously not an equilateral cone/triangle.

Any help most appreciated.

Byron Rogers said:
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.##

Ray Vickson said:
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?

Byron Rogers said:
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.

## 1. How do you calculate the volume of an ellipse cone?

To calculate the volume of an ellipse cone, you can use the formula V = (1/3)πr2h, where r is the radius of the base of the cone and h is the height of the cone. However, since an ellipse cone has an elliptical base, the radius (r) will be the average of the two semi-major and semi-minor axes (r = (a+b)/2).

## 2. What is the difference between an ellipse cone and a regular cone?

An ellipse cone has an elliptical base, while a regular cone has a circular base. This means that the formula for calculating the volume of an ellipse cone is slightly different from that of a regular cone.

## 3. Can you use the same formula to calculate the volume of a pyramid?

No, the formula for calculating the volume of an ellipse cone cannot be used to calculate the volume of a pyramid. Pyramids have a different shape and thus require a different formula for volume calculation.

## 4. What is the importance of calculating the volume of an ellipse cone?

Calculating the volume of an ellipse cone is important in various fields such as mathematics, engineering, and physics. It allows us to determine the amount of space that an ellipse cone occupies, which is useful in various real-world applications.

## 5. Can the volume of an ellipse cone be negative?

No, the volume of an ellipse cone cannot be negative. Volume is a measure of the amount of space occupied by an object, and it is always a positive value. If the calculated volume of an ellipse cone is negative, it is likely an error in the calculation.

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