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I am trying to calculate the network distance from a point to the nearest poly line feature as pointed in the image below.

Is there any ArcGIS Pro tool that can calculate network distance?

enter image description here

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ArcGIS Pro has Network Analyst. Inside the Network Analyst extension you can perform tasks like "Routing" (find the shortest path between two points across a network).

You'll need a streets network, constructed as a network dataset to use the Network solver.

This tutorial is a great starting point. If you don't have your own streets featureclass / network dataset, you can make use of ArcGIS Online services to perform this workflow. Keep in mind, this requires credits. The instructions to use the ArcGIS Online solver are included in the previously mentioned tutorial.

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