One of the first geometric formulas we learn in plane geometry is that the area of a triangle is:
Area of a Triangle = (1/2) * Base Width * Height.
So it is natural to wonder how this might generalize to pyramids in n-dimensional geometry. For instance, in 3-dimensions, the volume of a pyramid is:
Volume of Pyramid = (1/3) * Base Area * Height.
The same formula actually holds for a cone in 3-dimensions as well. Traditionally, one thinks of a cone as an object whose base B is circular, but in fact when the base is any shape, mathematicians still call the object a cone over B, and the formula above still holds for a 3-dimensional cone over any shape B. In general, the cone over any n-dimensional object B is the (n+1)-dimensional object formed by taking a point P outside the n-dimensional hyperplane spanned by B and taking the union of all the line segments from P to points in B. And the volume of such a cone is:
Volume of a Cone over B = (1/n+1) * Volume of B * Height.
Here, the “Height” is the distance from P from the hyperplane spanned by B.
Presentation Suggestions:
Although, the concept of volume in n-dimensional space is something that students sometimes find difficult to comprehend, one may motivate the idea by explaining that the notion of volume is basically a way to quantify the “size” of a set in n-dimensional space in a way that is translation-invariant.
The Math Behind the Fact:
The factor of (1/n+1) is probably the most interesting part about this formula. One way to see where this comes from is to use calculus. Consider a thin slice of the Cone over B, cut by planes parallel to the base B. This slice has cross-sectional volume that is a similar figure to B, except that in each dimension it has been scaled by (x/H). So, if the thickness of the slice is represented by dx, the volume of this slice is represented by:
(Volume of B)*(x/H)n dx,
and integrating this from x=0 to x=H yields the formula above. Moreover, we can see that the factor (1/n+1) emerges from integrating the xn in the expression above!
How to Cite this Page:
Su, Francis E., et al. “Volume of a Cone in N Dimensions.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.
Fun Fact suggested by:
Francis Su