diff --git a/django/contrib/gis/geos/prepared.py b/django/contrib/gis/geos/prepared.py
index d5b531bb430..90a3f4df214 100644
--- a/django/contrib/gis/geos/prepared.py
+++ b/django/contrib/gis/geos/prepared.py
@@ -1,6 +1,7 @@
-from django.contrib.gis.geos.base import GEOSBase
-from django.contrib.gis.geos.geometry import GEOSGeometry
-from django.contrib.gis.geos.prototypes import prepared as capi
+from .base import GEOSBase
+from .geometry import GEOSGeometry
+from .libgeos import geos_version_info
+from .prototypes import prepared as capi
 
 
 class PreparedGeometry(GEOSBase):
@@ -35,3 +36,30 @@ class PreparedGeometry(GEOSBase):
 
     def intersects(self, other):
         return capi.prepared_intersects(self.ptr, other.ptr)
+
+    # Added in GEOS 3.3:
+
+    def crosses(self, other):
+        if geos_version_info()['version'] < '3.3.0':
+            raise GEOSException("crosses on prepared geometries requires GEOS >= 3.3.0")
+        return capi.prepared_crosses(self.ptr, other.ptr)
+
+    def disjoint(self, other):
+        if geos_version_info()['version'] < '3.3.0':
+            raise GEOSException("disjoint on prepared geometries requires GEOS >= 3.3.0")
+        return capi.prepared_disjoint(self.ptr, other.ptr)
+
+    def overlaps(self, other):
+        if geos_version_info()['version'] < '3.3.0':
+            raise GEOSException("overlaps on prepared geometries requires GEOS >= 3.3.0")
+        return capi.prepared_overlaps(self.ptr, other.ptr)
+
+    def touches(self, other):
+        if geos_version_info()['version'] < '3.3.0':
+            raise GEOSException("touches on prepared geometries requires GEOS >= 3.3.0")
+        return capi.prepared_touches(self.ptr, other.ptr)
+
+    def within(self, other):
+        if geos_version_info()['version'] < '3.3.0':
+            raise GEOSException("within on prepared geometries requires GEOS >= 3.3.0")
+        return capi.prepared_within(self.ptr, other.ptr)
diff --git a/django/contrib/gis/geos/prototypes/prepared.py b/django/contrib/gis/geos/prototypes/prepared.py
index eef141a2f7a..ced56395b0c 100644
--- a/django/contrib/gis/geos/prototypes/prepared.py
+++ b/django/contrib/gis/geos/prototypes/prepared.py
@@ -1,5 +1,5 @@
 from ctypes import c_char
-from django.contrib.gis.geos.libgeos import GEOM_PTR, PREPGEOM_PTR
+from django.contrib.gis.geos.libgeos import GEOM_PTR, PREPGEOM_PTR, geos_version_info
 from django.contrib.gis.geos.prototypes.errcheck import check_predicate
 from django.contrib.gis.geos.prototypes.threadsafe import GEOSFunc
 
@@ -24,3 +24,10 @@ prepared_contains = prepared_predicate(GEOSFunc('GEOSPreparedContains'))
 prepared_contains_properly = prepared_predicate(GEOSFunc('GEOSPreparedContainsProperly'))
 prepared_covers = prepared_predicate(GEOSFunc('GEOSPreparedCovers'))
 prepared_intersects = prepared_predicate(GEOSFunc('GEOSPreparedIntersects'))
+
+if geos_version_info()['version'] > '3.3.0':
+    prepared_crosses = prepared_predicate(GEOSFunc('GEOSPreparedCrosses'))
+    prepared_disjoint = prepared_predicate(GEOSFunc('GEOSPreparedDisjoint'))
+    prepared_overlaps = prepared_predicate(GEOSFunc('GEOSPreparedOverlaps'))
+    prepared_touches = prepared_predicate(GEOSFunc('GEOSPreparedTouches'))
+    prepared_within = prepared_predicate(GEOSFunc('GEOSPreparedWithin'))
diff --git a/django/contrib/gis/geos/tests/test_geos.py b/django/contrib/gis/geos/tests/test_geos.py
index 8356343839c..4c38c9bfcbe 100644
--- a/django/contrib/gis/geos/tests/test_geos.py
+++ b/django/contrib/gis/geos/tests/test_geos.py
@@ -1032,6 +1032,16 @@ class GEOSTest(unittest.TestCase, TestDataMixin):
             self.assertEqual(mpoly.intersects(pnt), prep.intersects(pnt))
             self.assertEqual(c, prep.covers(pnt))
 
+        if geos_version_info()['version'] > '3.3.0':
+            self.assertTrue(prep.crosses(fromstr('LINESTRING(1 1, 15 15)')))
+            self.assertTrue(prep.disjoint(Point(-5, -5)))
+            poly = Polygon(((-1, -1), (1, 1), (1, 0), (-1, -1)))
+            self.assertTrue(prep.overlaps(poly))
+            poly = Polygon(((-5, 0), (-5, 5), (0, 5), (-5, 0)))
+            self.assertTrue(prep.touches(poly))
+            poly = Polygon(((-1, -1), (-1, 11), (11, 11), (11, -1), (-1, -1)))
+            self.assertTrue(prep.within(poly))
+
         # Original geometry deletion should not crash the prepared one (#21662)
         del mpoly
         self.assertTrue(prep.covers(Point(5, 5)))
diff --git a/docs/ref/contrib/gis/geos.txt b/docs/ref/contrib/gis/geos.txt
index a1f92b31c7f..769c2cdd32a 100644
--- a/docs/ref/contrib/gis/geos.txt
+++ b/docs/ref/contrib/gis/geos.txt
@@ -527,8 +527,8 @@ a :class:`Polygon`).
 
 Returns a GEOS ``PreparedGeometry`` for the contents of this geometry.
 ``PreparedGeometry`` objects are optimized for the contains, intersects,
-and covers operations.  Refer to the :ref:`prepared-geometries` documentation
-for more information.
+covers, crosses, disjoint, overlaps, touches and within operations. Refer to
+the :ref:`prepared-geometries` documentation for more information.
 
 .. attribute:: GEOSGeometry.srs
 
@@ -746,8 +746,48 @@ For example::
 
   .. method:: covers(other)
 
+  .. method:: crosses(other)
+
+    .. versionadded:: 1.7
+
+    .. note::
+
+       GEOS 3.3 is *required* to use this predicate.
+
+  .. method:: disjoint(other)
+
+    .. versionadded:: 1.7
+
+    .. note::
+
+       GEOS 3.3 is *required* to use this predicate.
+
   .. method:: intersects(other)
 
+  .. method:: overlaps(other)
+
+    .. versionadded:: 1.7
+
+    .. note::
+
+       GEOS 3.3 is *required* to use this predicate.
+
+  .. method:: touches(other)
+
+    .. versionadded:: 1.7
+
+    .. note::
+
+       GEOS 3.3 is *required* to use this predicate.
+
+  .. method:: within(other)
+
+    .. versionadded:: 1.7
+
+    .. note::
+
+       GEOS 3.3 is *required* to use this predicate.
+
 Geometry Factories
 ==================
 
diff --git a/docs/releases/1.7.txt b/docs/releases/1.7.txt
index 21449b4e7c8..106de20ec15 100644
--- a/docs/releases/1.7.txt
+++ b/docs/releases/1.7.txt
@@ -211,6 +211,10 @@ Minor features
 * The default OpenLayers library version included in widgets has been updated
   from 2.11 to 2.13.
 
+* Prepared geometries now also support the ``crosses``, ``disjoint``,
+  ``overlaps``, ``touches`` and ``within`` predicates, if GEOS 3.3 or later is
+  installed.
+
 :mod:`django.contrib.messages`
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^