diff --git a/modules/juce_graphics/geometry/juce_EdgeTable.cpp b/modules/juce_graphics/geometry/juce_EdgeTable.cpp
index 5caa8bf130..43caa9ece6 100644
--- a/modules/juce_graphics/geometry/juce_EdgeTable.cpp
+++ b/modules/juce_graphics/geometry/juce_EdgeTable.cpp
@@ -32,8 +32,12 @@ const int juce_edgeTableDefaultEdgesPerLine = 32;
 //==============================================================================
 EdgeTable::EdgeTable (Rectangle<int> area, const Path& path, const AffineTransform& transform)
    : bounds (area),
-     maxEdgesPerLine (juce_edgeTableDefaultEdgesPerLine),
-     lineStrideElements (juce_edgeTableDefaultEdgesPerLine * 2 + 1)
+     // this is a very vague heuristic to make a rough guess at a good table size
+     // for a given path, such that it's big enough to mostly avoid remapping, but also
+     // not so big that it's wasteful for simple paths.
+     maxEdgesPerLine (jmax (juce_edgeTableDefaultEdgesPerLine / 2,
+                            4 * (int) std::sqrt (path.numElements))),
+     lineStrideElements (maxEdgesPerLine * 2 + 1)
 {
     allocate();
     int* t = table;
@@ -404,7 +408,7 @@ void EdgeTable::remapTableForNumEdges (const int newNumEdgesPerLine)
 
 inline void EdgeTable::remapWithExtraSpace (int numPoints)
 {
-    remapTableForNumEdges (numPoints + jmax (juce_edgeTableDefaultEdgesPerLine, numPoints / 2));
+    remapTableForNumEdges (numPoints * 2);
     jassert (numPoints < maxEdgesPerLine);
 }
 
diff --git a/modules/juce_graphics/geometry/juce_Path.h b/modules/juce_graphics/geometry/juce_Path.h
index c868b1ddde..eda4f2d5b1 100644
--- a/modules/juce_graphics/geometry/juce_Path.h
+++ b/modules/juce_graphics/geometry/juce_Path.h
@@ -781,9 +781,7 @@ public:
     void loadPathFromData (const void* data, size_t numberOfBytes);
 
     /** Stores the path by writing it out to a stream.
-
         After writing out a path, you can reload it using loadPathFromStream().
-
         @see loadPathFromStream, loadPathFromData
     */
     void writePathToStream (OutputStream& destination) const;
@@ -803,6 +801,8 @@ private:
     //==============================================================================
     friend class PathFlatteningIterator;
     friend class Path::Iterator;
+    friend class EdgeTable;
+
     ArrayAllocationBase<float, DummyCriticalSection> data;
     size_t numElements = 0;