diff --git a/src/sage/geometry/polyhedron/base.py b/src/sage/geometry/polyhedron/base.py
index 48f86e50fec..e20b8b70745 100644
--- a/src/sage/geometry/polyhedron/base.py
+++ b/src/sage/geometry/polyhedron/base.py
@@ -3605,7 +3605,7 @@ def volume(self, engine='auto', **kwds):
             2.37764129...
             sage: P5 = polytopes.regular_polygon(5, base_ring=QQ)
             sage: P5.volume()   # rational approximation
-            3387471714099766473500515673753476175274812279494567801326487870013/1424719417220622426561086640229666223984528142237277803327699435400
+            143675742936485206271005807482349119225365261915467953640852591/60427846494832899490396166935397049960830782710733164218307960
             sage: _.n()
             2.37764129...
 
diff --git a/src/sage/geometry/polyhedron/library.py b/src/sage/geometry/polyhedron/library.py
index b6e79bb13b1..30618f34f56 100644
--- a/src/sage/geometry/polyhedron/library.py
+++ b/src/sage/geometry/polyhedron/library.py
@@ -19,7 +19,7 @@
 ########################################################################
 
 
-from sage.rings.all import Integer, QQ, ZZ, RDF
+from sage.rings.all import Integer, RR, QQ, ZZ, RDF
 from sage.matrix.constructor import matrix
 from sage.modules.free_module_element import vector
 from sage.combinat.permutation import Permutations
@@ -121,27 +121,41 @@ def _pfunc(i,j,perm):
     @rename_keyword(deprecation=11634, field='base_ring')
     def regular_polygon(self, n, base_ring=QQ):
         """
-        Return a regular polygon with n vertices.  Over the rational
+        Return a regular polygon with `n` vertices.  Over the rational
         field the vertices may not be exact.
 
         INPUT:
 
         - ``n`` -- a positive integer, the number of vertices.
 
-        - ``field`` -- either ``QQ`` or ``RDF``.
+        - ``base_ring`` -- a ring in which the coordinates will lie.
 
         EXAMPLES::
 
             sage: octagon = polytopes.regular_polygon(8)
             sage: len(octagon.vertices())
             8
+            sage: polytopes.regular_polygon(3).vertices()
+            (A vertex at (-125283617/144665060, -500399958596723/1000799917193445),
+             A vertex at (0, 1),
+             A vertex at (94875313/109552575, -1000799917193444/2001599834386889))
+            sage: polytopes.regular_polygon(3, base_ring=RealField(100)).vertices()
+            (A vertex at (0.00000000000000000000000000000, 1.0000000000000000000000000000),
+             A vertex at (0.86602540378443864676372317075, -0.50000000000000000000000000000),
+             A vertex at (-0.86602540378443864676372317076, -0.50000000000000000000000000000))
+            sage: polytopes.regular_polygon(3, base_ring=RealField(10)).vertices()
+            (A vertex at (0.00, 1.0),
+             A vertex at (0.87, -0.50),
+             A vertex at (-0.86, -0.50))
         """
-        npi = 3.14159265359
+        try:
+            omega = 2*base_ring.pi()/n
+        except AttributeError:
+            omega = 2*RR.pi()/n
         verts = []
         for i in range(n):
-            t = 2*npi*i/n
-            verts.append([sin(t),cos(t)])
-        verts = [[base_ring(RDF(x)) for x in y] for y in verts]
+            t = omega*i
+            verts.append([base_ring(t.sin()), base_ring(t.cos())])
         return Polyhedron(vertices=verts, base_ring=base_ring)