diff --git a/include/wx/headerctrl.h b/include/wx/headerctrl.h
index f2270bf78e..8d677eec67 100644
--- a/include/wx/headerctrl.h
+++ b/include/wx/headerctrl.h
@@ -159,6 +159,12 @@ public:
     // compute column title width
     int GetColumnTitleWidth(const wxHeaderColumn& col);
 
+    // compute column title width for the column with the given index
+    int GetColumnTitleWidth(unsigned int idx)
+    {
+        return GetColumnTitleWidth(GetColumn(idx));
+    }
+
     // implementation only from now on
     // -------------------------------
 
diff --git a/interface/wx/headerctrl.h b/interface/wx/headerctrl.h
index d8045cf6de..3980a9a01e 100644
--- a/interface/wx/headerctrl.h
+++ b/interface/wx/headerctrl.h
@@ -404,6 +404,16 @@ public:
      */
     int GetColumnTitleWidth(const wxHeaderColumn& col);
 
+    /**
+        Returns width needed for the column with the given index.
+
+        This is just a convenient wrapper for the overload taking
+        wxHeaderColumn.
+
+        @since 3.1.3
+     */
+    int GetColumnTitleWidth(unsigned int idx);
+
 protected:
     /**
         Method to be implemented by the derived classes to return the
diff --git a/src/generic/grid.cpp b/src/generic/grid.cpp
index 77700ba016..e25a2ea050 100644
--- a/src/generic/grid.cpp
+++ b/src/generic/grid.cpp
@@ -9384,13 +9384,30 @@ wxGrid::AutoSizeColOrRow(int colOrRow, bool setAsMin, wxGridDirection direction)
 
     if ( column )
     {
-        dc.GetMultiLineTextExtent( GetColLabelValue(colOrRow), &w, &h );
-        if ( GetColLabelTextOrientation() == wxVERTICAL )
-            w = h;
+        if ( m_useNativeHeader )
+        {
+            w = GetGridColHeader()->GetColumnTitleWidth(colOrRow);
+            h = 0;
+        }
+        else
+        {
+            dc.GetMultiLineTextExtent( GetColLabelValue(colOrRow), &w, &h );
+            if ( GetColLabelTextOrientation() == wxVERTICAL )
+                w = h;
+
+            // leave some space around text
+            if ( w )
+                w += 10;
+        }
     }
     else
+    {
         dc.GetMultiLineTextExtent( GetRowLabelValue(colOrRow), &w, &h );
 
+        if ( h )
+            h += 6;
+    }
+
     extent = column ? w : h;
     if ( extent > extentMax )
         extentMax = extent;
@@ -9401,14 +9418,6 @@ wxGrid::AutoSizeColOrRow(int colOrRow, bool setAsMin, wxGridDirection direction)
         // than default extent but != 0, it's OK)
         extentMax = column ? m_defaultColWidth : m_defaultRowHeight;
     }
-    else
-    {
-        if ( column )
-            // leave some space around text
-            extentMax += 10;
-        else
-            extentMax += 6;
-    }
 
     if ( column )
     {
diff --git a/src/msw/renderer.cpp b/src/msw/renderer.cpp
index 2b1a5470af..9372afbd29 100644
--- a/src/msw/renderer.cpp
+++ b/src/msw/renderer.cpp
@@ -590,9 +590,10 @@ int wxRendererMSW::GetHeaderButtonHeight(wxWindow * win)
     return Header_Layout(hwndHeader, &hdl) ? wp.cy : DEFAULT_HEIGHT;
 }
 
-int wxRendererMSW::GetHeaderButtonMargin(wxWindow *WXUNUSED(win))
+int wxRendererMSW::GetHeaderButtonMargin(wxWindow *win)
 {
-    return 10;
+    // The native control seems to use 3*SM_CXEDGE margins on each size.
+    return 6*wxGetSystemMetrics(SM_CXEDGE, win);
 }
 
 // ============================================================================