diff --git a/core/src/test/java/org/elasticsearch/common/geo/GeoDistanceTests.java b/core/src/test/java/org/elasticsearch/common/geo/GeoDistanceTests.java
index 6624a91a6c2..c2e62277da7 100644
--- a/core/src/test/java/org/elasticsearch/common/geo/GeoDistanceTests.java
+++ b/core/src/test/java/org/elasticsearch/common/geo/GeoDistanceTests.java
@@ -81,29 +81,62 @@ public class GeoDistanceTests extends ESTestCase {
         assertThat(GeoUtils.rectangleContainsPoint(box, 0, -178), equalTo(false));
     }
 
-    /**
-     * The old plane calculation in 1.x/2.x incorrectly computed the plane distance in decimal degrees. This test is
-     * well intended but bogus. todo: fix w/ new plane distance calculation
-     * note: plane distance error varies by latitude so the test will need to correctly estimate expected error
-     */
-    @AwaitsFix(bugUrl = "old plane calculation incorrectly computed everything in degrees. fix this bogus test")
-    public void testArcDistanceVsPlaneInEllipsis() {
-        GeoPoint centre = new GeoPoint(48.8534100, 2.3488000);
-        GeoPoint northernPoint = new GeoPoint(48.8801108681, 2.35152032666);
-        GeoPoint westernPoint = new GeoPoint(48.85265, 2.308896);
+    private static double arcDistance(GeoPoint p1, GeoPoint p2) {
+        return GeoDistance.ARC.calculate(p1.lat(), p1.lon(), p2.lat(), p2.lon(), DistanceUnit.METERS);
+    }
 
-        // With GeoDistance.ARC both the northern and western points are within the 4km range
-        assertThat(GeoDistance.ARC.calculate(centre.lat(), centre.lon(), northernPoint.lat(),
-                northernPoint.lon(), DistanceUnit.KILOMETERS), lessThan(4D));
-        assertThat(GeoDistance.ARC.calculate(centre.lat(), centre.lon(), westernPoint.lat(),
-                westernPoint.lon(), DistanceUnit.KILOMETERS), lessThan(4D));
+    private static double planeDistance(GeoPoint p1, GeoPoint p2) {
+        return GeoDistance.PLANE.calculate(p1.lat(), p1.lon(), p2.lat(), p2.lon(), DistanceUnit.METERS);
+    }
 
-        // With GeoDistance.PLANE, only the northern point is within the 4km range,
-        // the western point is outside of the range due to the simple math it employs,
-        // meaning results will appear elliptical
-        assertThat(GeoDistance.PLANE.calculate(centre.lat(), centre.lon(), northernPoint.lat(),
-                northernPoint.lon(), DistanceUnit.KILOMETERS), lessThan(4D));
-        assertThat(GeoDistance.PLANE.calculate(centre.lat(), centre.lon(), westernPoint.lat(),
-                westernPoint.lon(), DistanceUnit.KILOMETERS), greaterThan(4D));
+    public void testArcDistanceVsPlane() {
+        // sameLongitude and sameLatitude are both 90 degrees away from basePoint along great circles
+        final GeoPoint basePoint = new GeoPoint(45, 90);
+        final GeoPoint sameLongitude = new GeoPoint(-45, 90);
+        final GeoPoint sameLatitude = new GeoPoint(45, -90);
+
+        double sameLongitudeArcDistance = arcDistance(basePoint, sameLongitude);
+        double sameLatitudeArcDistance = arcDistance(basePoint, sameLatitude);
+        double sameLongitudePlaneDistance = planeDistance(basePoint, sameLongitude);
+        double sameLatitudePlaneDistance = planeDistance(basePoint, sameLatitude);
+
+        // GeoDistance.PLANE measures the distance along a straight line in
+        // (lat, long) space so agrees with GeoDistance.ARC along a line of
+        // constant longitude but takes a longer route if there is east/west
+        // movement.
+
+        assertThat("Arc and plane should agree on sameLongitude",
+            Math.abs(sameLongitudeArcDistance - sameLongitudePlaneDistance), lessThan(0.001));
+
+        assertThat("Arc and plane should disagree on sameLatitude (by >4000km)",
+            sameLatitudePlaneDistance - sameLatitudeArcDistance, greaterThan(4.0e6));
+
+        // GeoDistance.ARC calculates the great circle distance (on a sphere) so these should agree as they're both 90 degrees
+        assertThat("Arc distances should agree", Math.abs(sameLongitudeArcDistance - sameLatitudeArcDistance), lessThan(0.001));
+    }
+
+    public void testArcDistanceVsPlaneAccuracy() {
+        // These points only differ by a few degrees so the calculation methods
+        // should match more closely. Check that the deviation is small enough,
+        // but not too small.
+
+        // The biggest deviations are away from the equator and the poles so pick a suitably troublesome latitude.
+        GeoPoint basePoint = new GeoPoint(randomDoubleBetween(30.0, 60.0, true), randomDoubleBetween(-180.0, 180.0, true));
+        GeoPoint sameLongitude = new GeoPoint(randomDoubleBetween(-90.0, 90.0, true), basePoint.lon());
+        GeoPoint sameLatitude = new GeoPoint(basePoint.lat(), basePoint.lon() + randomDoubleBetween(4.0, 10.0, true));
+
+        double sameLongitudeArcDistance = arcDistance(basePoint, sameLongitude);
+        double sameLatitudeArcDistance = arcDistance(basePoint, sameLatitude);
+        double sameLongitudePlaneDistance = planeDistance(basePoint, sameLongitude);
+        double sameLatitudePlaneDistance = planeDistance(basePoint, sameLatitude);
+
+        assertThat("Arc and plane should agree [" + basePoint + "] to [" + sameLongitude + "] (within 1cm)",
+            Math.abs(sameLongitudeArcDistance - sameLongitudePlaneDistance), lessThan(0.01));
+
+        assertThat("Arc and plane should very roughly agree [" + basePoint + "] to [" + sameLatitude + "]",
+            sameLatitudePlaneDistance - sameLatitudeArcDistance, lessThan(600.0));
+
+        assertThat("Arc and plane should disagree by some margin [" + basePoint + "] to [" + sameLatitude + "]",
+            sameLatitudePlaneDistance - sameLatitudeArcDistance, greaterThan(15.0));
     }
 }