Both phrases are used to indicate the interior location or position of something within a larger space or context.  The choice between "inside of which" and "inside of which" depends on the formality and context of your communication.

How to use "inside which" and "inside of which"

Both phrases usually appear after the noun of which the interior is being described.

What sets it apart is the software inside which uses real-time kinematic technology to enhance the precision of the GPS measurements. - TechCrunch
In this quote from TechCrunch, you can see that the inside which is being used to describe the usage of "real-time kinematic technology" in the interior of a software. 
There was a small crater next to the damaged armored personnel carrier, the inside of which was stained with blood. - The New York Times - Magazine
In this quote from The New York Times, inside of which directs the reader's attention to the blood that is inside of the personnel carrier.

The use of inside which vs. inside of which

From below you can see that the use of "inside of which" has decreased, while the use of "inside which" has increased since the 1800s.

Inside Which

Use this in formal writing, such as academic papers or official documents. It is more common in British English and is preferred when a more formal tone is required.

Examples from the web

What sets it apart is the software inside, which uses real-time kinematic technology to enhance the precision of the GPS measurements. - TechCrunch
For example, the team let me make a couple punches using gloves with sensors inside, which were created for the mixed martial arts league UFC. - TechCrunch
The channel is characterized by roughness features on the inside, which can be analyzed using optical microscopy techniques. - Journal of Materials Processing Technology
I wandered inside, which is outside, which is inside. - The New Yorker

More example sentences

  1. Inside which box did you find the missing puzzle piece?
  2. The treasure chest, inside which glimmered a collection of rare coins, was a sight to behold.
  3. Could you tell me the suitcase inside which I packed my favorite clothes?
  4. Inside which container did you store the holiday decorations?
  5. The cave, inside which the adventurers discovered an underground river, was a natural wonder.
  6. Inside which car should I place the groceries before heading home?
  7. The building, inside which the art exhibition was held, was beautifully decorated.
  8. Inside which room can I find the documents you mentioned?
  9. The forest, inside which a tranquil pond could be found, was a peaceful retreat.
  10. Can you please point out the cabinet inside which the fine china is stored?

Inside of Which

This is well-suited for casual conversations and informal writing. It is also more commonly used in American English.

Examples from the web

There was a small crater next to the damaged armored personnel carrier, the inside of which was stained with blood. - The New York Times - Magazine
Lee's device placed a number of fuses — small lead tubes capped with hemispherical heads of very thin metal, inside of which there were small glass vials of sulfuric acid — in a metal casing filled with gunpowder. - The New York Times
Thought to be more than 1,000 years old, it has a hollow trunk of about 12 metres, the inside of which was once used as a tearoom with door and roof built in. - The Guardian - Travel
Just a few, though, will head left before taking a sharp right along the underground walkway and disappear through a badly lit revolving door, on the inside of which is a police guard. - The Guardian

More example sentences

  1. The mysterious locked chest had a hidden compartment inside of which was a valuable map.
  2. The old library had a secret passage inside of which a treasure trove of ancient books was hidden.
  3. The antique armoire had a hidden drawer inside of which precious family heirlooms were kept.
  4. The cave had a narrow tunnel inside of which the explorers found a hidden chamber filled with ancient artifacts.
  5. The attic was cluttered with old boxes, inside of which were memories of years gone by.
  6. The abandoned house had a basement inside of which strange symbols were discovered on the walls.
  7. The vintage trunk had a hidden compartment inside of which old love letters were preserved.
  8. The old mansion had a dusty attic inside of which a forgotten painting by a famous artist was found.
  9. The old theater had a secret room inside of which actors used to prepare for their performances.
  10. The forest had a dense thicket inside of which a hidden clearing offered a tranquil spot for picnics.

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Tricks to Remember the Difference:
A simple trick to remember the difference is to consider the level of formality. "Inside which" is more formal, while "inside of which" is less formal and often used in spoken language.

Synonyms

  • Within Which: A synonym for both "inside which" and "inside of which," often used to convey the same idea with a slightly different structure.
    • Within the grand library, you will find a treasure trove of ancient manuscripts and rare books, within which lies the history of generations
  • In Which: Another synonym that can replace both phrases when discussing something contained within something else.
    • In the bustling city, she found a peaceful park, in which she could escape the noise and chaos of urban life.

Common Idioms

  1. In the Heart of: An idiom often used to describe a location or situation that is deeply within a particular area or context.
  2. Within the Confines of: Used when discussing something located or occurring within specific boundaries or limits.

Summary

In summary, "inside which" and "inside of which" are two variations that differ in formality and context. Use "inside which" in formal or written contexts, especially in British English, and opt for "inside of which" in casual conversations and informal writing, particularly in American English. Understanding this distinction will help you convey your message with the appropriate tone and style.


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References:

inside which or inside of which?
inside which or inside of which? - Which English form is more popular?