  We investigate geometric configurations of $\alpha$ ($^4$He nucleus) clusters
in the second $J^\pi=2^+$ state of $^{12}$C, which has been discussed as a
rotational band member of the second $0^+$ state, the Hoyle state. The ground
and excited $0^+$ and $2^+$ states are described by a three-$\alpha$ cluster
model. The three-body Schr\"odinger equation with orthogonality conditions is
accurately solved by the stochastic variational method with correlated Gaussian
basis functions. To analyse the structure of these resonant states in a
convenient form, we introduce a confining potential. The two-body density
distributions together with the spectroscopic information clarify the structure
of these states. We find that main configurations of both the second $0^+$ and
$2^+$ states are acute-angled triangle shapes originating from the
$^8$Be($0^+$)$+\alpha$ configuration. However, the $^8$Be$+\alpha$ components
in the second $2^+$ state become approximately 2/3 because the 8Be subsystem is
hard to excite, indicating that the state is not an ideal rigid rotational band
member of the Hoyle state.
