# 10.4: D- Volume of a Sphere in d Dimensions

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I will call the volume of a *d*-dimensional sphere, as a function of radius, *V _{d}*(

*r*). You know, of course, that

\[ V_{2}(r)=\pi r^{2}\]

(two-dimensional volume is commonly called “area”) and that

\[ V_{3}(r)=\frac{4}{3} \pi r^{3}.\]

But what is the formula for arbitrary *d*? There are a number of ways to find it. I will use induction on dimensionality *d*. That is, I will use the formula for *d* = 2 to find the formula for *d* = 3, the formula for *d* = 3 to find the formula for *d* = 4, and in general use the formula for *d* to find the formula for *d* + 1. This is not the most rigorous formal method to derive the formula, but it is very appealing and has much to recommend it.

To illustrate the process, I will begin with a well-known and easily visualized stage, namely deriving *V*_{3}(*r*) from *V*_{2}(*r*). Think of a 3-dimensional sphere (of radius *r*) as a stack of pancakes of various radii, but each with infinitesimal thickness dz. The pancake on the very bottom of the stack (*z* = −*r*) has zero radius. The one above it is slightly broader. They get broader and broader until we get to the middle of the stack (*z* = 0), where the pancake has radius *r*. The pancakes stacked still higher become smaller and smaller, until they vanish again at the top of the stack (*z* = +*r*). Because the equation for the sphere is

\[ x^{2}+y^{2}+z^{2}=r^{2},\]

the radius of the pancake at height *z*_{0} is

\[ \sqrt{r^{2}-z_{0}^{2}}.\]

This whole process shows that

\[ V_{3}(r)=\int_{-r}^{+r} d z V_{2}\left(\sqrt{r^{2}-z^{2}}\right).\]

It is easy to check this integral against the known result for *V*_{3}(*r*):

\[ V_{3}(r)=\int_{-r}^{+r} d z \pi\left(r^{2}-z^{2}\right)\]

\[ =\pi\left[r^{2} z-\frac{1}{3} z^{3}\right]_{-r}^{+r}\]

\[ =\pi\left[2 r^{3}-\frac{2}{3} r^{3}\right]\]

\[ =\frac{4}{3} \pi r^{3}.\]

So we haven’t gone wrong yet.

Now, how to derive *V*_{4}(*r*) from *V*_{3}(*r*)? This requires a more vivid imagination. Last time we started with a two-dimensional disk of radius *r*_{0} in (*x, y*) space and thickened it a bit into the third dimension (*z*) to form a pancake of three-dimensional volume *dz* *V*_{2}(*r*_{0}). Stacking an infinite number of such pancakes in the *z* direction, from *z* = −*r* to *z* = +*r*, gave us a three-dimensional sphere. Now we begin with a three-dimensional sphere of radius *r*_{0} in (*w, x, y*) space and thicken it a bit into the fourth dimension (z) to form a thin four-dimensional pancake of four-dimensional volume *dz* *V*_{3}(*r*_{0}). Stacking an infinite number of such pancakes in the *z* direction, from *z* = −*r* to *z* = +*r*, gives a four-dimensional sphere. Because the equation for the four-sphere is

\[ w^{2}+x^{2}+y^{2}+z^{2}=r^{2},\]

the radius of the three-dimensional sphere at height *z*_{0} is

\[ \sqrt{r^{2}-z_{0}^{2}},\]

and the volume of the four-sphere is

\[ V_{4}(r)=\int_{-r}^{+r} d z V_{3}\left(\sqrt{r^{2}-z^{2}}\right).\]

In general, the volume of a (*d* + 1)-sphere is

\[ V_{d+1}(r)=\int_{-r}^{+r} d z V_{d}\left(\sqrt{r^{2}-z^{2}}\right).\]

If we guess that the formula for *V _{d}*(

*r*) takes the form

\[ V_{d}(r)=C_{d} r^{d}\]

(which is certainly true for two and three dimensions, and which is reasonable from dimensional analysis), then

\[ V_{d+1}(r)=\int_{-r}^{+r} d z C_{d}\left(r^{2}-z^{2}\right)^{d / 2}\]

\[ =\int_{-1}^{+1} r d u C_{d}\left(r^{2}-r^{2} u^{2}\right)^{d / 2}\]

\[ =r^{d+1} C_{d} \int_{-1}^{+1} d u\left(1-u^{2}\right)^{d / 2}.\]

This proves our guess and gives us a recursive formula for *C _{d}*:

\[ C_{d+1}=C_{d} \int_{-1}^{+1} d u\left(1-u^{2}\right)^{d / 2}.\]

The problem below shows how to build this recursive chain up from *C*_{2} = π to

\[ C_{d}=\frac{\pi^{d / 2}}{\Gamma\left(\frac{d}{2}+1\right)}=\frac{\pi^{d / 2}}{(d / 2) !}.\]

Thus the volume of a *d*-dimensional sphere of radius *r* is

\[ V_{d}(r)=\frac{\pi^{d / 2}}{(d / 2) !} r^{d}.\]

*D.1 (I) Problem: Volume of a d-dimensional sphere *

Before attempting this problem, you should read the material concerning beta functions in an applied mathematics textbook, such as George Arfken’s *Mathematical Methods for Physicists* or Mary Boas’s *Mathematical Methods in the Physical Sciences*. (Or in the Digital Library of Mathematical Functions.)

a. Show that

\[ \int_{-1}^{+1}\left(1-u^{2}\right)^{d / 2} d u=B\left(\frac{1}{2}, \frac{d}{2}+1\right).\]

b. Use

\[ B(p, q)=\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}\]

and *C*_{2} = π to conclude that

\[ C_{d}=\frac{\pi^{d / 2}}{\Gamma\left(\frac{d}{2}+1\right)}.\]

*D.2 (I) Problem: Volume of a d-dimensional ellipse *

Show that the volume of the d-dimensional ellipse described by the equation

\[ \left(\frac{x_{1}}{a_{1}}\right)^{2}+\left(\frac{x_{2}}{a_{2}}\right)^{2}+\left(\frac{x_{3}}{a_{3}}\right)^{2}+\cdots+\left(\frac{x_{d}}{a_{d}}\right)^{2}=1\]

is

\[ V_{d}(r)=\frac{\pi^{d / 2}}{(d / 2) !} a_{1} a_{2} a_{3} \cdots a_{d}.\]