diff --git a/xarray/core/coordinates.py b/xarray/core/coordinates.py
index 28cbc5b7622..0886e07b41b 100644
--- a/xarray/core/coordinates.py
+++ b/xarray/core/coordinates.py
@@ -24,6 +24,7 @@
 from xarray.core.types import DataVars, Self, T_DataArray, T_Xarray
 from xarray.core.utils import (
     Frozen,
+    FrozenMappingWarningOnValuesAccess,
     ReprObject,
     either_dict_or_kwargs,
     emit_user_level_warning,
@@ -111,12 +112,13 @@ def to_dataset(self) -> Dataset:
         raise NotImplementedError()
 
     def to_index(self, ordered_dims: Sequence[Hashable] | None = None) -> pd.Index:
-        """Convert all index coordinates into a :py:class:`pandas.Index`.
+        """Convert all index dimension coordinates into a :py:class:`pandas.Index`.
 
         Parameters
         ----------
         ordered_dims : sequence of hashable, optional
-            Possibly reordered version of this object's dimensions indicating
+            Possibly reordered version of this object's dimensions (or the full dimensions
+            of it's corresponding Dataset, DataArray or DataTree object) indicating
             the order in which dimensions should appear on the result.
 
         Returns
@@ -124,14 +126,14 @@ def to_index(self, ordered_dims: Sequence[Hashable] | None = None) -> pd.Index:
         pandas.Index
             Index subclass corresponding to the outer-product of all dimension
             coordinates. This will be a MultiIndex if this object is has more
-            than more dimension.
+            than one dimension.
         """
         if ordered_dims is None:
-            ordered_dims = list(self.dims)
-        elif set(ordered_dims) != set(self.dims):
+            ordered_dims = list(self._data.dims)
+        elif set(ordered_dims) != set(self._data.dims):
             raise ValueError(
                 "ordered_dims must match dims, but does not: "
-                f"{ordered_dims} vs {self.dims}"
+                f"{ordered_dims} vs {tuple(self._data.dims)}"
             )
 
         if len(ordered_dims) == 0:
@@ -741,8 +743,8 @@ def _names(self) -> set[Hashable]:
 
     @property
     def dims(self) -> Frozen[Hashable, int]:
-        # deliberately display all dims, not just those on coordinate variables - see https://github.com/pydata/xarray/issues/9466
-        return self._data.dims
+        dims = calculate_dimensions(self.variables)
+        return FrozenMappingWarningOnValuesAccess(dims)
 
     @property
     def dtypes(self) -> Frozen[Hashable, np.dtype]:
@@ -851,8 +853,8 @@ def _names(self) -> set[Hashable]:
 
     @property
     def dims(self) -> Frozen[Hashable, int]:
-        # deliberately display all dims, not just those on coordinate variables - see https://github.com/pydata/xarray/issues/9466
-        return Frozen(self._data.dims)
+        dims = calculate_dimensions(self.variables)
+        return FrozenMappingWarningOnValuesAccess(dims)
 
     @property
     def dtypes(self) -> Frozen[Hashable, np.dtype]:
@@ -942,7 +944,8 @@ def __init__(self, dataarray: T_DataArray) -> None:
 
     @property
     def dims(self) -> tuple[Hashable, ...]:
-        return self._data.dims
+        dims = calculate_dimensions(self._data._coords)
+        return tuple(dims)
 
     @property
     def dtypes(self) -> Frozen[Hashable, np.dtype]:
@@ -967,7 +970,9 @@ def _update_coords(
         self, coords: dict[Hashable, Variable], indexes: dict[Hashable, Index]
     ) -> None:
         validate_dataarray_coords(
-            self._data.shape, Coordinates._construct_direct(coords, indexes), self.dims
+            self._data.shape,
+            Coordinates._construct_direct(coords, indexes),
+            self._data.dims,
         )
 
         self._data._coords = coords
diff --git a/xarray/tests/test_dataarray.py b/xarray/tests/test_dataarray.py
index 71c11626e1e..333f397d95d 100644
--- a/xarray/tests/test_dataarray.py
+++ b/xarray/tests/test_dataarray.py
@@ -1522,6 +1522,12 @@ def test_coords(self) -> None:
             self.mda["level_1"] = ("x", np.arange(4))
             self.mda.coords["level_1"] = ("x", np.arange(4))
 
+    def test_coords_dims(self) -> None:
+        da = DataArray(
+            [[1.0, 2.0], [3.0, 4.0]], coords={"a": ("x", [0, 1])}, dims=("x", "y")
+        )
+        assert da.coords.dims == ("x",)
+
     def test_coords_to_index(self) -> None:
         da = DataArray(np.zeros((2, 3)), [("x", [1, 2]), ("y", list("abc"))])
 
diff --git a/xarray/tests/test_dataset.py b/xarray/tests/test_dataset.py
index 8e4b09fbfeb..f456f8e83b0 100644
--- a/xarray/tests/test_dataset.py
+++ b/xarray/tests/test_dataset.py
@@ -892,6 +892,11 @@ def test_coords_properties(self) -> None:
             "b": np.dtype("int64"),
         }
 
+    def test_coords_dims(self) -> None:
+        # https://github.com/pydata/xarray/issues/9466
+        ds_no_coord = Dataset(data_vars={"a": ("x", [0, 1])})
+        assert not len(ds_no_coord.coords.dims)
+
     def test_coords_modify(self) -> None:
         data = Dataset(
             {
diff --git a/xarray/tests/test_datatree.py b/xarray/tests/test_datatree.py
index 2bf079a7cbd..186233a00a2 100644
--- a/xarray/tests/test_datatree.py
+++ b/xarray/tests/test_datatree.py
@@ -666,6 +666,18 @@ def test_properties(self) -> None:
             "b": np.dtype("int64"),
         }
 
+    def test_dims(self) -> None:
+        # https://github.com/pydata/xarray/issues/9466
+        ds = Dataset(
+            data_vars={
+                "foo": (["x", "y"], np.random.randn(2, 3)),
+            },
+        )
+        dt = DataTree(dataset=ds)
+        dt["child"] = DataTree()
+
+        assert not len(dt.coords.dims)
+
     def test_modify(self) -> None:
         ds = Dataset(
             data_vars={