What method does the Measure tool in the ArcGIS Online map viewer use to create measurements?

Answer

The Measure tool in ArcGIS Online uses the Geodesic method, which takes into account the curvature of the earth to create accurate measurements.

Note:
Geodesic measures the shortest line between any two points on the earth's surface based on a spheroid (ellipsoid). 

Distortions are expected when measuring distances or areas in higher latitudes when (by default ArcGIS in Online) maps are projected in the Web Mercator coordinate system. However, measurements are relatively accurate when using the Measure tool in ArcGIS Online because the tool uses a method that honors the ellipsoid of the coordinate system of the basemap.

Note:
When measuring in a data frame with a projected coordinate system, the default measurement type is Planar, which does not take into account the curvature of the earth.