Volume of Gr(2,4) Hello
I was wondering if anybody can direct me to a paper or a book regarding the volume of $Gr(2,4) $ or generic complex Grassmanian manifolds of order $k$. My own heuristic method seems not to work! It is based on the adaption of the same procedure one has to follow for finding the volume of complex projective spaces $\mathbb{C}P^n$ using Hopf fibration $\mathbb{C}P^n\cong S^{2n+1}/S^1$. Here the volume can be roughly given by dividing the volume of $2n+1-$sphere by volume of $S^1$. Therefore in analogy with this example, we can estimate the volume of $Gr(k,n)$ by dividing the volume of $U(n)$ by that of $U(n-k) \times U(k)$ which gives me $12\pi^4 r^{16}$ for $Gr(2,4)$ where $r$ is the radius of $S^1$ and I don't like it because $Gr(2,4) $ is $8$ dimensional! 
Thanks in Advance
AB
 A: The volume of a Grassmanian can be computed using Wirtinger's theorem: 

The volume of a $p$-dimensional complex submanifold $S$ of a complex Hermitian manifold $(X,\omega)$  is 
$$
\frac{1}{p\!}\int_S\omega^p.
$$

If $X=\mathbb{CP}^N$ the integral is equal to the degree of $S$ times the volume of $X$. Thus up to normalization factors, the volume of the Grassmanian $Gr(k,n)$ is its degree in the Plücker embedding $$Gr(k,n)\subset \mathbb{CP}^N, N=\binom{n}{k}-1.$$
A: Check section 9.1.2 of these notes  There I compute the volumes of real Grassmannians. A similar computation works in the complex case.
Update  Using the  description $\mathrm{Gr}\;(k, N)\cong U(N/U(k)\times U(N-k)$ and a bi-invariant metric on $U(N)$, this induces  bi-invaraint metrics on $U(k),U(N-k)\subset U(n)$ and an invariant metric on $\mathrm{Gr}(k,N)$. The volume  of $\mathrm{Gr}(k,N)$ with respect to this metric is
$$ {\rm vol} \mathrm{Gr}(k, N)= \frac{ {\rm vol}\; U(N)}{{\rm vol}\; U(k)\cdot {\rm vol}\; U(N-k)}. $$
The volume of a compact Lie group $G$   with respect to a bi-invariant  metric $g$ was computed by I.G. Macdonald, 

The volume of a compact Lie group, Invent. Math. 56(1980), no. 2, 93–95.

For the Lie group $U(n)$ this takes the form
$$ {\rm vol}\; U(n)=\frac{1}{(2P_n)^2(2\pi)^n}\times {\rm vol}\; T^n\times \prod_{k=1}^n {\rm vol}\;S^{2k-1}, $$
where ${\rm vol}\; T^n$  denotes the volume of the maximal torus of $U(n)$ equipped with the induced bi-invariant metric,  and $P_n$ is the product  of the lengths of the positive roots of $U(n)$.
